Sz.-Nagy Centennial Conference

June 24-28, 2013,    Szeged, Hungary

Welcome Plenary speakers Invited speakers Important Dates Registration Venue Participants Abstract submission

List of abstracts as of 06/30/2013, 21:14 CEST

Spectral properties of Toeplitz operators on the unit ball and on the unit sphere

Universität des Saarlandes

This is a report on joint work with Zineb Akkar. We consider Toeplitz operators on the Hardy and weighted Bergman Hilbert spaces of the unit sphere respectively of the unit ball in $\mathbb{C}^N$. Various aspects of the interplay between local and global properties of the symbols and local and global spectral properties of the corresponding Toeplitz operators are investigated. A local version of the spectral inclusion theorem of Davie and Jewell [D-J] is proved. Using some recent results of Quiroga-Barranco and Vasilevski [Q-V1,Q-V2], we describe some commutative $C^*$-subalgebras of the Toeplitz algebra for $N\geq 2$. The method of McDonald [McD] to compute the essential spectrum of Toeplitz operators with certain piecewise continuous symbols is extended to a larger class of symbols including examples where the surface measure of the set of discontinuity points is strictly positive.

[D-J]
Davie, A.M. and Jewell, N.P., Toeplitz operators in several complex variables, J. Functional Analysis 26 (1977), 356-368.

[McD]
McDonald, G., Toeplitz operators on the ball with piecewise continuous symbols, Illinois Journal of Mathematics 23 (1979), 286-294.

[Q-V1]
Quiroga-Barranco, R. and Vasilevski, N., Commutative C*-Algebras of Toeplitz operators on the unit ball, I. Bargmann-type transforms and spectral representations of Toeplitz operators, Integr. equ. oper. theory 59 (2007), no. 3, 379-419.

[Q-V2]
Quiroga-Barranco, R. and Vasilevski, N., Commutative C*-Algebras of Toeplitz operators on the unit ball, II. Geometry of the level sets of symbols, Integr. equ. oper. theory 60 (2008), no. 1, 89-132.

On certain multivariable subnormal weighted shifts and their duals

Indian Institute of Technology Bombay

To every subnormal $m$-variable weighted shift $S$ (with bounded positive weights) corresponds a positive Reinhardt measure $\mu$ supported on a compact Reinhardt subset of ${\mathbb C}^m$. We show that, for $m \geq 2$, the dimensions of the $1$st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual ${\tilde S}$ are different if a certain radial function happens to be integrable with respect to $\mu$ (which is indeed the case with many classical examples); in particular, $S$ cannot in that case be similar to ${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual. (This is joint work with Pramod Patil).

Linear stability of the constant 1-D gas flow with respect to the initial value and permanent source produced time harmonic perturbation

West University of Timisoara

The stability of the null solution of the linearized Euler equations in four different functional frameworks is analyzed. It is shown that the methods and results depend on the meaning of the concepts solution and small, which are specific for each framework. As concerns the stability with respect to the initial value perturbation it is found that: in three of the frameworks the solution is stable, but in one of them it is unstable; in one of the frameworks the solution is stable, although the set of the exponential growth rates of the solutions is the whole real axis; in two of the frameworks the exponential growth rates of the solutions are equal to zero, but only in one of them the Briggs-Bers stability analysis can be applied; in all of the considered frameworks the problem is well posed in sense of Hadamard. As concerns the stability with respect to permanent source produced time harmonic perturbations, it is found: in three of the frameworks the problem is well posed, but the null solution is unstable; in one of the frameworks the problem is ill posed; just in one of the frameworks the Briggs-Bers stability analysis can be applied. The above results can be useful in a better understanding of some apparently strange results obtained in different published papers concerning the sound propagation in a gas flowing through a lined duct.

Joint work with Agneta M. Balint, West University of Timisoara.

Weighted Bergman spaces: characteristic functions and functional models for $n$-hypercontraction operators

Virginia Tech

The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators combined with the Beurling-Lax theorem establishes a correspondence between any two of four kinds of objects: shift-invariant subspaces, operator-valued inner functions, conservative discrete-time input/state/output linear systems, and $C_{\cdot 0}$ Hilbert-space contraction operators. We present an analogue of all these ideas in the context of weighted Bergman spaces and $n$-hypercontraction operators. We also discuss partial extensions of these results to more general weighted Hardy spaces and directions for future work. This is joint work with Vladimir Bolotnikov of the College of William & Mary.

PDE approximation of large systems of differential equations

Loránd Eötvös University, Institute of Mathematics

A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin bondary condition. Using the theory of differential operators with Wentzell boundary conditions and similar theories, we give estimates on the order of approximation. The theory is demonstrated on a voter model where the Fourier method applied to the PDE seems to be of great advantage.

Intersection theory and invariant subspaces

Indiana University

It is known that the description of the possible invariant subspaces of an operator of class $C_0$ or, more simply, of an algebraic operator involves the Littlewood-Richardson rule. This rule was originally designed to describe the structure constants of the representation ring of $GL(n)$, which also coincide with (some of) the structure constants of the intersection ring of a Grassmannian. It is natural to ask whether a direct connection could be found between intersection theory on Grassmannians and invariant subspaces on the other. We answer this question in the affirmative.

This is joint work with W. S. Li and K. Dykema, and it uses essentially work of the present three authors with B. Collins and D. Timotin.

Reflexive sets of operators

University of Ljubljana

Let ${\mathcal T}$ be a set of bounded linear operators from a complex Banach space $X$ to a Banach space $Y$. An operator $S$ from $X$ to $Y$ is said to be locally in ${\mathcal T}$ if $Sx$ belongs to the closure of the set $\{ Tx;~T\in{\mathcal T}\}$ for every $x\in X$. Let ${\rm Ref}\,{\mathcal T}$ denote the set of all operators which are locally in ${\mathcal T}$. It is obvious that ${\mathcal T} \subseteq {\rm Ref}\,{\mathcal T}$. If ${\rm Ref}\,{\mathcal T}={\mathcal T}$, then ${\mathcal T}$ is said to be a reflexive set. Since the concept of reflexivity is so simple and natural it was studied during the last few decades by many authors.

We will discuss a few questions related to the reflexivity and to the stronger notion of hyperreflexivity. Some results about algebraically reflexive sets will be presented, as well.

Unbounded quasinormal composition operators in $L^2$-spaces

University of Agriculture in Krakow

Let $(X, \mathcal{A}, \mu)$ be a $\sigma$-finite measure space and let $\phi$ be a nonsingular transformation of $X$. Then the map $C_\phi\colon L^2(\mu) \supseteq {\mathcal D}(C_\phi)\to L^2(\mu)$ given by \begin{align*} {\mathcal D}(C_\phi) = \{f \in L^2(\mu) \colon f \circ \phi \in L^2(\mu)\} \text{ and } C_\phi f = f \circ \phi \text{ for } f \in {\mathcal D}C_\phi), \end{align*} is well-defined and linear. Such an operator is called a composition operator.

Let $\mathcal{H}$ be a complex Hilbert space. A densely defined linear operator $A$ in $\mathcal{H}$ is said to be quasinormal if $A$ is closed and $U |A| \subseteq |A|U$, where $A=U|A|$ is the polar decomposition of $A$.

It was shown by R. Whitley that a bounded composition operator $C_\phi$ is quasinormal if and only if ${\mathsf h}_\phi = {\mathsf h}_\phi \circ \phi$ a.e. $[\mu]$, where ${\mathsf h}_{\phi^n}$ denotes the Radon-Nikodym derivative $d\mu\circ(\phi^n)^{-1}/d\mu$, $n = 0, 1, 2, \ldots$. On the other hand, A. Lambert noticed that if $C_{\phi}$ is a bounded quasinormal composition operator with a surjective symbol $\phi$, then the Radon-Nikodym derivatives ${\mathsf h}_{\phi^n}$ have the following multiplicative property: ${\mathsf h}_{\phi^n} = {\mathsf h}_{\phi}^n$ a.e. $[\mu]$ for $n = 0, 1, 2, \ldots$. We extend these results to the case of unbounded operators. Namely, we show that they in fact characterize quasinormality of densely defined composition operators in $L^2$-spaces.

This talk is based on joint work with Zenon Jan Jabłoński, Il Bong Jung and Jan Stochel.

[b-j-j-sC]
P. Budzyński, Z. J. Jabłoński, I. B. Jung, J. Stochel, On unbounded composition operators in $L^2$-spaces, Ann. Mat. Pura Appl. doi:10.1007/s10231-012-0296-4.

[b-j-j-sE]
P. Budzyński, Z. J. Jabłoński, I. B. Jung, J. Stochel, A multiplicative property characterizes quasinormal composition operators in $L^2$-spaces, submitted for publication.

One-sided invertibility, corona problems and applications to Toeplitz operators

Technical University of Lisbon

When is a matricial Toeplitz operator Fredholm equivalent to a scalar Toeplitz operator (meaning that either they are both Fredholm with the same index, or they are both non-Fredholm)? When does it have Coburn's property? Can left-invertibility of an $n\times m$ ($m\leq n$) matrix over a unital commutative ring be studied in terms of an associated scalar corona problem? These apparently independent questions are addressed taking an algebraic approach which moreover provides a good illustration of how the study of Toeplitz operators knits together different areas of mathematics.

The Sz.Nagy-Foias model and factorization in certain preduals

Chevreau, Bernard

University Bordeaux 1

The use of the Sz.Nagy-Foias functional model for factorization in preduals goes back to the beginning of dual algebra theory in the eighties with, among the most signicant ones, works of Bercovici, Foias, Kérchy, Pearcy and Sz.-Nagy himself (cf. references in [BFP]).

In the nineties in [H] (and in several other papers), the author offered a tentative approach to the invariant subspace problem for contractions with spectral radius one via very lengthy and technical factorizations in preduals of certain weak$\star$-closed subspaces of $H^\infty$ (the Banach algebra of bounded analytic functions in the open unit disk), also based on the functional model of a contraction. Though this approach has not (so far) being successfull (in terms of existence of invariant subspaces) it seems to be of independant interest. In the talk, after reviewing the early use of the functional model in this area, we will present a considerably simplified version of the approximation scheme of [H] with emphasize on the role of the "special" vectors.

[BFP]
Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Math., No. 56, AMS, Providence, RI, 1985.

[H]
Subspaces of $H^\infty$ and the study of contractions with spectral radius one, Rev. Roumaine de Math. Pures et Appl., 41 (1996), 1-2, 51-82.

Truncated versions of moment problems via positive definiteness on semigroups

Jagiellonian University

Applying a general theorem on positive definite extensions of functions on subsets of $*$-semigroups we obtain numerous results concerning truncated versions of classical moment problems such as the complex moment problem, the multidimensional trigonometric moment problem or the two-sided complex moment problem. What is more, employing the general theorem one can obtain criteria for subnormality of an unbounded operator as well as for existence of joint unitary power dilation of several Hilbert space operators.

This talk is based on a joint work with J. Stochel and F.H. Szafraniec.

Fredholm properties for a class of Toeplitz operators with symbols with a gap around zero

Diogo, Cristina

University Institute of Lisbon

The study of finite interval convolution equations is closely related to the study of Toeplitz operators with symbols of the form $$G=\left[ \begin{array}{cc} e^{-i\lambda\xi} & 0 \\ g(\xi) & e^{i\lambda\xi} \\ \end{array}% \right]\,, \lambda>0\,, g\in L_\infty (\mathbb{R})\,, \xi\in\mathbb{R}.$$

In this talk, we consider triangular matrix functions with a non-diagonal entry $g$ of the form $g=a_-e^{-i\beta\xi}+a_+e^{i\nu\xi}\,,$ with $a_\pm\in H_\infty (\mathbb{C}^\pm)$ and $\nu,\beta >0.$ We show that, in this case, solutions to the Riemann-Hilbert problem $$Gh_+=h_-\,, \hspace{0.2cm} \text{with}\hspace{0.2cm} \,h_\pm\in (H_\infty (\mathbb{C}^\pm))^2,$$ can be explicitly determined and, moreover, provide conditions for Fredholmness and invertibility of Toeplitz operators with symbols $G$ of the above form.

Generalized Canonical Models

Texas A&M University

Although the canonical model which Sz.-Nagy and Foias constructed for all contraction operators on Hilbert spaces, it often provides definitive results on structure when the model multiplicity is finite. For example, although there is a model for the Bergman shift, it has infinite multiplicity, and hence provided little insight into the fine structure of that operator.

One way to broaden the "finite-multiplicity" class is to consider canonical models built using other reproducing kernel Hilbert spaces of holomorphic functions. In this talk we explore the outlines of such an approach, casting it in the language of Hilbert modules and including the multivariate case as well.

On subnormality of weighted shift operators

Jagiellonian University

Weighted shifts on directed trees is a relatively new class of operators which serves as an interesting research area with many open problems [Sto]. This class provides fascinating examples of operators which fulfill surprising properties not satisfied by operators from well known classes. In the talk we present an example of a subnormal operator with a dense domain and with such property that the domain of its $n$-th power is not dense.

Subnormal operators constitute a wide and intensively studied class of operators defined in Hilbert spaces, however, there does not exist a simple and efficient subnormality criterion, which could be used in order to verify whether a given weighted shift on a tree is subnormal.

On the basis of Ando's construction [Ando], we provide sufficient and necessary conditions for subnormality and the subnormal extension in the case of a bounded weighted shift operator on a directed tree with one branching point which can be used in particular cases to verify that a weighted shift operator is not subnormal.

[Ando]
T. Ando. Matrices of normal extensions of subnormal operators, Acta Sci. Math. (Szeged) 24(1963), 91--96.

[Sto]
Z.Jabłoński, I.B.Jung, J.Stochel. Weighted shifts on directed trees, Memoirs of the AMS 216 (2012).

Wold decompositions and stationary dilations for multivariate random fields distributions

"Aurel Vlaicu" University

The extension of a continuous multivariate (normal Hilbert module valued) random field on ${\mathbb R}^d$ to a corresponding module valued distribution on ${\mathbb R}^d$ leads to a special stochastic mapping (called multivariate random field distribution), where the index set differs from the time parameter set. For such stochastic mappings determinism, non-determinism, Wold type decomposition into deterministic and purely non-deterministic part, operator stationarity and dilation to operator stationary multivariate random field distribution are considered.

Dilation of operator valued module maps on some generalized Hilbert modules

West University of Timioşara

This is a joint work with Păstorel Gaşpar.

Lately the technique of Hilbert $C^*$-modules represents an important tool in the development of operator theory. Such a technique is also very useful in the study of multivariate stochastic processes and their prediction. In this paper some generalizations of Hilbert $C^*$-modules such as Hilbert and Finsler locally $C^*$-modules, as well as Hilbert modules over the algebra ${\mathcal B}({\mathcal X})$ of all bounded linear operators on the Banach space ${\mathcal X}$, with ${\mathcal B}({\mathcal X}, {\mathcal X}^*)$-valued inner product, are considered. In such frames general Sz.-Nagy-type dilation theorems of positive definite module maps are presented. Finally these theorems are applied to obtain Naimark type dilations of corresponding semi-spectral measures.

Positive operators arising asymptotically from contractions

Gehér, György Pál

Bolyai Institute

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{B}(\mathcal{H})$ stand for the C*-algebra of bounded, linear operators on $\mathcal{H}$. The operator $T\in\mathcal{B}(\mathcal{H})$ is a contraction if $\|T\|\leq 1$. Let us consider the sequence $\{T^{*n}T^n\}_{n=1}^\infty$ of positive operators, which is decreasing, so it has a limit in the strong operator topology (SOT): $A_T := \lim_{n\to\infty}T^{*n}T^n.$ We say that $A_T$ arises asymptotically from $T$, or $A_T$ is the asymptotic limit of $T$. In the case when this convergence holds in norm, we say that $A_T$ arises asymptotically from $T$ in uniform convergence or $A_T$ is the uniform asymptotic limit of $T$. We recall that $A_T^{1/2}$ acts as an intertwining mapping in a canonical realization of the so called unitary and isometric asymptote of the contraction $T$, which is a very efficient tool in the theory of Hilbert space contractions.
The main goal of the talk is to give a complete characterization of those positive operators that are asymptotic limits of contractions. It turns out that in this case the positive operator arises asymptotically from a contraction in uniform convergence. We also investigate some connections between contractions which have the same asymptotic limit. We shall use the isometric asymptote and the Sz.-Nagy—Foias functional calculus for this purpose.

On subordinated bounded holomorphic $C_0$-semigroups

Gomilko, Oleksandr

Nicolas Copernicus University

We give a positive answer to the the following question from p. 63 [Rob]: If $-A$ generates a (sectorially) bounded holomorphic $C_0$-semigroup and $\psi$ is a Bernstein function, then whether $-\psi(A)$ generates a bounded holomorphic $C_0$-semigroup too ? A partial positive answer to this question was given in Proposition 7.4 in [Laub], where it was proved that $-\psi(A)$ generates a bounded holomorphic $C_0$-semigroup if $-A$ is the generator of a bounded holomorphic $C_0$-semigroup of angle greater than $\pi/4$. We also prove that if $\psi$ is a Bernstein function then a bounded $C_0$-semigroup $(e^{-t\psi(A)})_{t\ge 0}$ satisfies the Yosida condition $\sup_{t\in (0,1]}\,\|t\psi(A)e^{-t\psi(A)}\|<\infty,$ whenever $(e^{-t A})_{t\ge 0}$ does.

This is a joint work with Yu. Tomilov.

[Rob]
A. Kishimoto and D. W. Robinson, Subordinate semigroups and order properties, J. Austral. Math. Soc. Ser. A 31 (1981), 59--76.

[Laub]
C. Berg, K. Boyadzhiev and R. deLaubenfels, Generation of generators of holomorphic semigroups, J. Austral. Math. Soc. Ser. A 55 (1993), 246--269.

Non-hyponormal operators which generate Stieltjes moment sequences

Jagiellonian University

A linear operator $S$ in a complex Hilbert space $\mathcal{H}$ for which the set $\mathcal{D}^\infty(S)$ of its $C^\infty$-vectors is dense in $\mathcal{H}$ and $\{\|S^n f\|^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every $f \in \mathcal{D}^\infty(S)$ is said to generate Stieltjes moment sequences. The celebrated Lambert's theorem states that a bounded linear operator $S$ on $\mathcal{H}$ is subnormal if and only if $S$ generates Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal (and thus non-subnormal) operator $S$ which generates Stieltjes moment sequences. What is more, $\mathcal{D}^\infty(S)$ is a core of any power $S^n$ of $S$. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of indeterminate Stieltjes moment sequences. As a consequence, it is shown that there exists a non-hyponormal composition operator in an $L^2$-space (over a $\sigma$-finite measure space) which is injective and paranormal, and which generates Stieltjes moment sequences. Moreover, it satisfies any of the three equivalent Lambert's conditions which completely characterize the subnormality of bounded composition operators.

The talk is based on:

[1]
Z. J. Jablonski, I. B. Jung, J. Stochel, A non-hyponormal operator generating Stieltjes moment sequences, Journal of Functional Analysis 262 (2012), 3946-3980.

[2]
P. Budzynski, Z. J. Jablonski, I. B. Jung, J. Stochel, On unbounded composition operators in $L^2$-spaces, to appear in Annali di Matematica Pura ed Applicata.

Hermitian Operators on $H^\infty_E$ and $S^\infty_K$

Jamison, James

University of Memphis

A complete characterization of bounded and unbounded norm hermitian operators on $H^{\infty}_E$ is given for the case when $E$ is a complex Banach space with trivial multiplier algebra . As a consequence,the bi-circular projections on $H^{\infty}_E$ are determined. We also characterize a subclass of hermitian operators on $S^{\infty}_K$ for K a complex Hilbert space.

Hyper-Invariant subspaces for some compact perturbation of a diagonal operator

Klaja, Hubert

Université Lille 1

The hyper-invariant subspace problem for a bounded operator $T$ acting on a separable complex Hilbert space $H$, such that $T \ne \lambda I$, is the question whether there exist a non trivial closed subspace $M$, such that for every bounded operator $S$ commuting with $T$, $S(M) \subset M$. If $D$ is a diagonal operator and $K$ is a compact operator, it is still unknown in general whether $D+K$ has an hyper-invariant subspace.

In this talk we will discuss the existence of non trivial hyper-invariant subspaces for some operators of the form $D+K$.

On the unitary part of isometries in commuting, completely non doubly commuting pairs

Jagiellonian University

We describe a unitary extension of any isometry in a commuting, completely non doubly commuting pair of isometries. Precisely we show that Hilbert space of such an extension is a linear span of subspaces reducing the extension to bilateral shifts.

The coauthors of the presented results are Zbigniew Burdak, Patryk Pagacz and Marek Słociński.

On Biquasitriangular and ${\cal C}_{00}$-Contractions

Catholic University of Rio de Janeiro

If $T$ is a Hilbert-space contraction, then the sequence ${\{T^{*n}T^n\}}$ converges strongly to a nonnegative contraction $A$. If $A$ is a projection, then the contraction $T$ is said to be asymptotically partially isometric. A contraction $T$ is of $($Nagy-Foias$)$ class ${\cal C}_{0{\cdot}}$ if $A$ is null, and of $($Nagy-Foias$)$ class ${\cal C}_{00}$ if both $T$ and its adjoint $T^*$ are of class ${\cal C}_{0{\cdot}}$. An operator $T$ on a separable Hilbert space is quasitriangular if there is a sequence $\{P_n\}$ of finite-rank projections that converges strongly to the identity operator $I$ and ${\{(I-P_n)TP_n\}}$ converges uniformly to the null operator. An operator $T$ is biquasitriangular if both $T$ and $T^*$ are quasitriangular. We discuss the question: are biquasitriangular contractions asymptotically partially isometric? What is behind this question is that a positive answer to it yields a positive answer to the following classical open question: does a contraction not in ${\cal C}_{00}$ have a nontrivial invariant subspace? Indeed, if every biquasitriangular contraction is asymptotically partially isometric, then every contraction not in ${\cal C}_{00}$ has a nontrivial invariant subspace.

The continuation problem for hermitian functions and spectral measures of some differential operators

TU Vienna

We consider the close connection between the set of all spectral measures of a two-dimensional canonical system, a Sturm-Liouville operator or a string, and the continuations of a hermitian function such that a certain kernel is positive definite.

Extremal decompositions of variances

Léka, Zoltán

MTA Rényi Alfréd Institute

In this talk we study convex sums of noncommutative covariance matrices of operators acting on finite-dimensional Hilbert spaces. We obtain that rank-$1$ projections determine the covariance matrix if it is low dimensional. We also present an estimate on the rank in the general case.

Hyperreflexivity and masa bimodule projections

University College Dublin

This talk concerns ongoing joint work with Ivan Todorov and Georgios Eleftherakis.
Let $H$ be a separable Hilbert space and let $A$ be a masa in $B(H)$. We will discuss hyperreflexivity properties of subspaces of $B(H)$ of the form $U+\mathrm{Ran}(\Phi)$ where $U$ is a hyperreflexive $A$-bimodule in $B(H)$ and $\Phi$ is a normal, idempotent $A$-bimodule map.

Local and global liftings of analytic families of idempotents in Banach algebras

A. Renyi Inst. of Math., Hungar. Acad. Sci.

This is a joint work with Bernard Aupetit, Mostafa Mbekhta, and Jaroslav Zemánek. Generalizing results of our earlier paper, we investigate the following question. Let $\pi(\lambda) : A \rightarrow B$ be an analytic family of surjective homomorphisms between two Banach algebras, and $q(\lambda)$ an analytic family of idempotents in $B$. We want to find an analytic family $p(\lambda)$ of idempotents in $A$, lifting $q(\lambda)$, i.e., such that $\pi(\lambda)p(\lambda) = q(\lambda)$, under hypotheses of the type that the elements of\/ $\mathrm{Ker}\, \pi(\lambda)$ have small spectra. For spectra which do not disconnect $\mathbf{C}$ we obtain a local lifting theorem. For real analytic families of surjective $^*$-homomorphisms (for continuous involutions) and self-adjoint idempotents we obtain a local lifting theorem, for totally disconnected spectra. We obtain a global lifting theorem if the spectra of the elements in $\mathrm{Ker}\, \pi(\lambda)$ are $\{0\}$, both in the analytic case, and, for $^*$-algebras (with continuous involutions) and self-adjoint idempotents, in the real analytic case. Here even an at most countably infinite set of mutually orthogonal analytic families of idempotents can be lifted to mutually orthogonal analytic families of idempotents. In the proofs, spectral theory is combined with complex analysis and general topology, and even a connection with potential theory is mentioned.

B. Aupetit, E. Makai, Jr., M. Mbekhta, J. Zemánek, The connected components of the idempotents in the Calkin algebra, and their liftings in: Operator Theory and Banach Algebras, Conf. Proc., Rabat (Morocco), April 1999, Eds. M. Chidami, R. Curto, M. Mbekhta, F.-H. Vasilescu, J. Zemánek, Theta, Bucharest, 2003, 23--30

B. Aupetit, E. Makai, Jr., M. Mbekhta, J. Zemánek, Local and global liftings of analytic families of idempotents in Banach algebras Acta Sci. Math. (Szeged), accepted

On the reproducing kernel thesis for operators in Bergman-type spaces

Clemson University

I will introduce a concept of Bergman-type spaces which incorporates the Bergman and the Bargmann-Fock spaces as prime examples and give several criteria that imply boundedness and compactness of operators on these spaces. Most of these results can be viewed as “reproducing kernel thesis” statements. Namely, these results show that many crucial properties of a given operator can be deduced just by looking at its behavior on the reproducing kernels in the space.

This talk is based on joint work with B. Wick.

On the reflexivity of subspaces of Toeplitz operators in simply connected regions

University of Agriculture in Krakow

The reflexivity and transitivity of subspaces of Toeplitz operators on the Hardy space over a Jordan region — a simply connected region in the complex plane with analytic Jordan curve as its boundary — can be studied. We show the dichotomic behavior (either reflexive or transitive) of these subspaces. It refers to the similar dichotomic behavior for subspaces of Toeplitz operators on the Hardy space over the unit disc.

M. B. Abrahamse, Toeplitz Operators in Multiply Connected Regions, American Journal of Mathematics, Vol. 96, No. 2, pp. 261-297 (1974).

E. A. Azoff, M. Ptak, A Dichotomy for Linear Spaces of Toeplitz Operators, J. Funct. Anal. 156, 411-428 (1998).

S. Brown, B. Chevreau and C. Pearcy, Contractions with rich spectrum have invariant subspaces, J. Operator Theory, 1 (1979), 123-136.

W. Młocek, M. Ptak, On the reflexivity of subspaces of Toeplitz operators on the Hardy space on the upper half—plane, Czechoslovak Math. J., to appear.

W. Rudin, Analytic functions of class $H^p$, Transactions, American Mathematical Society, 78 (1955), pp. 46-66.

D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511-517.

Reflexivity of automorphism groups and isometry groups of some operator structures

University of Debrecen

Reflexivity results we are interested in basically concern the question if a given class of transformations on a given structure is determined by its local actions. For a more precise formulation we need the following concept. Suppose that the class $\mathcal G$ of transformations has the following property. For any transformation $\phi$ on the underlying structure which coincides on any pair $a,b$ of points with an element $\phi_{a,b}$ of $\mathcal G$ (possibly depending on $a,b$) meaning that $\phi(a)=\phi_{a,b}(a),\phi(b)=\phi_{a,b}(b)$, it necessarily follows that $\phi$ belongs to the class $\mathcal G$. In this case we say that the class $\mathcal G$ of transformations is algebraically reflexive.

Let $H$ be a separable complex Hilbert space. In the first part of this talk we shall present joint results with F. Botelho and J. Jamison stating that the isometry group of the unitary group on $H$ with the uniform norm, that of the set of all positive definite operators equipped with the Thompson metric, and that of the general linear group on $H$ with the uniform norm are all algebraically reflexive. As consequences, reflexivity results for related automorphism groups will also be obtained. A result of that kind typically states that any transformation which behaves two-pointwise as an automorphism is in fact an automorphism.

In the second part we consider a different concept called bilocal *-automorphism. Let $\mathcal A$ be a *-algebra of operators acting on a separable complex Hilbert space $H$. The linear transformation $\phi:\mathcal A \to \mathcal A$ is said to be a bilocal *-automorphism if for every $A\in \mathcal A$ and $x\in H$ there is an algebra *-automorphism $\phi_{A,x}$ of $\mathcal A$ such that $\phi(A)x=\phi_{A,x}(A)x$. We show that if $H$ is infinite dimensional, then a linear transformation $\phi$ on the full operator algebra over $H$ is a bilocal *-automorphism if and only if it is a unital *-endomorphism.

On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over $\overline{\mathbf{R}}^m_+$

Móricz, Ferenc

University of Szeged

We investigate the regular convergence of the $m$-multiple series $$\sum^\infty_{j_1=0} \sum^\infty_{j_2=0} \ldots \sum^\infty_{j_m=0} \ c_{j_1, j_2, \ldots, j_m} \tag{*}$$ of complex numbers, where $m\ge 2$ is a fixed integer. We prove Fubini's theorem in the discrete setting as follows. If the multiple series ($*$) converges regularly, then its sum in Pringsheim's sense can also be computed by successive summation.

We introduce and investigate the regular convergence of the $m$-multiple integral $$\int^\infty_0 \int^\infty_0 \ldots \int^\infty_0 f(t_1, t_2, \ldots, t_m) dt_1 dt_2 \ldots dt_m, \tag{**}$$ where $f: \overline{\mathbf{R}}^m_+ \rightarrow \mathbf{C}$ is a locally integrable function in Lebesgue's sense over the closed nonnegative octant $\overline{\mathbf{R}}^m_+:= [0, \infty)^m$. Our main result is a generalized version of Fubini's theorem on successive integration formulated in as follows. If $f\in L^1_{\mathrm{loc}} (\overline{\mathbf{R}}^m_+)$, the multiple integral ($**$) converges regularly, and $m=p+q$, where $p$ and $q$ are positive integers, then the finite limit \begin{multline} \lim_{v_{p+1}, \ldots, v_m \rightarrow \infty} \int^{v_1}_{u_1} \int^{v_2}_{u_2} \ldots \int^{v_p}_{u_p} \int^{v_{p+1}}_0 \ldots \int^{v_m}_0 f(t_1, t_2, \ldots, t_m) dt_1 dt_2\ldots dt_m \\ =:J(u_1, v_1; u_2, v_2; \ldots; u_p, v_p), \quad 0\le u_k\le v_k<\infty, \ k=1,2,\ldots, p, \end{multline} exists uniformly in each of its variables, and the finite limit $$\lim_{v_1, v_2, \ldots, v_p\rightarrow \infty} J(0, v_1; 0, v_2; \ldots; 0, v_p)=I$$ also exists, where $I$ is the limit of the multiple integral ($**$) in Pringsheim's sense.

The main results of this talk were announced without proofs in the Comptes Rendus Sci. Paris.

Universal operators and universal $n$-tuples

Institute of Mathematics of the Academy of Sciences of the Czech Republic

Let $H$ be an infinite-dimensional Hilbert space. By a classical result of Caradus, every surjective operator $T\in B(H)$ with infinite-dimensional kernel is universal in the following sense: for each operator $S$ on a separable Hilbert space there exist a constant $c>0$ and a subspace $M\subset H$ invariant for $T$ such that the restriction $T|M$ is similar to $cS$.
We will discuss the connections of universal operators with the dilation theory and generalizations of the Caradus result for $n$-tuples of operators, both in commutative and non-commutative setting.

On contractions in Hilbert space

Budapest Univ. of Technology and Economics

Dedicated to the memory of Professor Béla Sz.-Nagy

In the first part of the paper we study the decompositions of a (bounded linear) operator similar to a normal operator in Hilbert space into the orthogonal sum of a normal (self-adjoint, unitary) part and of a part free of the given property, respectively. In the second part we investigate in a finite dimensional Hilbert space the minimal unitary power dilations (till the exponent $k$) of a contraction. We determine the general form of such dilations, examine their spectra, and the question of their isomorphy. The first step of the study here is also the decomposition of the contraction into unitary and completely non-unitary parts.

AMS Subject Classifications (2010): 47A10, 47A20, 47A30.

Functional calculus for diagonalizable matrices

Jagiellonian University

Let $f: D \to \mathbb{C}$ (where $D \subset \mathbb{C}$) be any function and $k > 1$ be an integer. For a diagonalizable $k \times k$ matrix $X$ whose all eigenvalues lie in $D$, the matrix $f[X]$ is well defined by the rule $f[X] = P(X)$ where $P$ is an arbitrary complex polynomial which coincides with $f$ on the spectrum of $X$. When $f$ and $k$ are fixed and $X$ runs over all normal matrices with respective spectra (i.e. contained in $D$), it is well-known that the function $X \mapsto f[X]$ is continuous if and only if $f$ is so. In this talk we will give a full answer to a similar problem of when the function $X \mapsto f[X]$ is continuous if $X$ runs over all diagonalizable $k \times k$ matrices with respective spectra. It turns out that the characterization strongly depends on $k$ and involves functions of class $C^{k-2}$ (provided $D$ is a subinterval of the real line). During the talk the following topics will be discussed on the assignment $X \mapsto f[X]$: uniform and locally uniform continuity for fixed $k$; continuity independent of $k$; continuity in infinite dimension, that is, when $X$ runs over all diagonalizable as well as all scalar (i.e. similar to normal) bounded operators on $\ell_2$ with respective spectra.

P. Niemiec, Functional calculus for diagonalizable matrices, Linear Multilinear Algebra (2013), doi: 10.1080/03081087.2013.777440 (on-line first).

Dilation on the Symmetrized Bidisc

Pal, Sourav

Ben-Gurion University of the Negev

For a contraction $P$ and a bounded commutant $S$ of $P$, we seek a solution $X$ of the operator equation $$S-S^*P=(I-P^*P)^{\frac{1}{2}}X(I-P^*P)^{\frac{1}{2}},$$ where $X$ is a bounded operator on $\overline{\mbox{Ran}}(I-P^*P)^{\frac{1}{2}}$ with numerical radius of $X$ being not greater than $1$. A pair of bounded operators $(S,P)$ which has the domain, the symmetrized bidisc, $$\Gamma=\{(z_1+z_2,z_1z_2): |z_1|\leq1,|z_2|\leq1\}\subseteq \mathbb C^2$$ as a spectral set, is called a $\Gamma$-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a $\Gamma$-contraction $(S,P)$. This allows us to construct an explicit $\Gamma$-isometric dilation of a $\Gamma$-contraction $(S,P)$. Hence it follows that the existence of solution to that operator equation is a necessary and sufficient for a commuting pair $(S,P)$ to be a $\Gamma$-contraction.

New Characterizations of WEP

University of Houston

We will present some new characterizations of C*-algebras with Lance's weak expectation property(WEP). One involves an extension of Ando's work on numerical ranges and the second is via Riesz interpolation in the order on self-adjoint operators. These new characterizations are motivated by Kirchberg's work that shows that Connes' embedding problem is equivalent to deciding if certain group C*-algebras are WEP.

Quantum expanders and geometry of operator spaces

Texas A&M University and Université Paris VI

We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications of geometric applications to the "local theory" of operator spaces. This allows us to provide sharp estimates for the growth of the multiplicity of $M_N$-spaces needed to represent (up to a constant $C>1$) the $M_N$-version of the $n$-dimensional operator Hilbert space $OH_n$ as a direct sum of copies of $M_N$. We show that, when $C$ is close to 1, this multiplicity grows as $\exp{\beta n N^2}$ for some constant $\beta>0$. The main idea is to identify quantum expanders with smooth points on the matricial analogue of the unit sphere, and to show that there are plenty of uniformly smooth points (more precisely as many as allowed by a soft metric entropy dimensional restriction). This generalizes to operator spaces a classical geometric result on $n$-dimensional Hilbert space (corresponding to $N=1$). Our work leads to identify a certain class of operator spaces that we call matricially subGaussian, that appears as a generalization of the “exact” ones. We will discuss the connection of this notion with “exactness” for operator spaces.

On string density at the origin

Pivovarchik, Vyacheslav

South-Ukrainian National Pedagogical University

Let two sequences of real numbers $\{\mu_k\}_{k=1}^{\infty}$ and $\{\lambda_k\}_{k=1}^{\infty}$ satisfy $$0<\mu_1<\lambda_1<\mu_2<\lambda_2<...$$ and let $L$ be a positive number. It is known that there exists a string with regular left end and with a nondecreasing mass distribution function $M(x)$ ($x\in [0,L)$, $M(0)=0$) for which the problem with the Neumann condition at $x=0$ and Dirichlet condition at $x=L$ (if the right end is regular) has the spectrum $\{\mu_k\}_{k=1}^{\infty}$, while the problem with the Dirichlet condition at $x=0$ and Dirichlet condition at $x=L$ has the spectrum $\{\lambda_k\}_{k=1}^{\infty}$.

Using results of [Kac] and [Kasahara] we find the value of $\alpha>0$ for which the limit $\lim\limits_{x\to +0}\frac{M(x)}{x^{\alpha}}$ exists and is finite. We prove that if in addition to interlacing $$\mu_k=\frac{\pi ^2(k-1/2)^2}{b^2}+O(k^{\beta}), \ \ \lambda_k=\frac{\pi ^2k^2}{b^2}+O(k^{\beta})$$ where $b>0$, $\beta\in [0,1)$ then $$\lim\limits_{x\to+0}\frac{M(x)}{x}= \frac{1}{L^2\mu_1}\mathop{\prod}\limits_{n=1}^{\infty}\frac{\lambda_n^2}{\mu_n \mu_{n+1}}.$$ We compare this result with results of [Barcilon] and [Shen]. Unlike in these papers our results are true even if $M(x)$ has no absolutely continuous component.

This talk is based on collaboration with I.S. Kac.

[Barcilon]
V. Barcilon, Explicit solution of the inverse problem for a vibrating string. J. Math. Anal. Appl. 93 (1983) 222-234

[Kac]
I.S. Kac, Generalization of an asymptotic formula of V.A. Marchenko for spectral functions of a second-order boundary value problem. Math. USSR Izvestija, Vol. 7 (1973), no. 2, 424--438

[Kasahara]
Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes. Japan J. Math. 1 (1975)

[Shen]
C.-L. Shen, On the Barcilon formula for the string equation with a piecewise continuous density function. Inverse Problems 21, (2005) 635--655

Automorphisms of spectral order

University of Agriculture in Krakow

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{E}(\mathcal{H})=\{ A\in\boldsymbol{B}_s(\mathcal{H})\colon 0\le A \le I\}$. Full characterization of automorphisms of $\mathcal{E}(\mathcal{H})$ with respect to spectral order (cf. [Olson], [p-s]) was given in the paper [ms] by L. Molnár and P. Šemrl.

Let $\kappa\in\mathbb{N}$. In this talk I am going to characterize every automorphism of the set $\mathcal{E}^\kappa(\mathcal{H})=\{\textbf{A}\in\mathcal{E}(\mathcal{H})^\kappa\colon \textbf{A} \mbox{ is commutative } \}$ with respect to the multidimensional spectral order.

[ms]
L. Molnár and P. Šemrl, Spectral order automorphisms of the spaces of Hilbert space effects and observables, Lett. Math. Phys., 80 (2007), 239-255.

[Olson]
M. P. Olson, The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc. Amer. Math. Soc. 28 (1971), 537-544.

[p-s]
A. Płaneta, J. Stochel, Spectral order for unbounded operator, J. Math. Anal. Appl., 389 (2012), 1029 - 1045.

Operator theory on noncommutative polydomains

The University of Texas at San Antonio

This is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs and extend it to a more general class of noncommutative polydomains ${\mathbf D}\subset B(H)^n$. Several aspects of the Sz.-Nagy—Foias theory of contractions are extended to our more general setting. Each polydomain ${\mathbf D}$ has a universal model ${\mathbf S}$ consisting of weighted shifts acting on a tensor product of full Fock spaces. We study the universal model ${\mathbf S}$, its joint invariant subspaces, and the representations of the universal operator algebras it generates: the polydomain algebra $A({\mathbf D})$, the Hardy algebra $F^\infty({\mathbf D})$, and the $C^*$-algebra $C^*({\mathbf D})$. Using noncommutative Berezin transforms, we provide a characterization of the Beurling type joint invariant subspaces under the universal model and develop an operator model theory and dilation theory for these noncommutative polydomains. We obtain a characterization of those elements in $\mathbf D$ which admit characteristic functions and prove that the characteristic function is a complete unitary invariant for the class of completely non-coisometric elements. The commutative case is also discussed.

On operators which are adjoint to each other

Popovici, Dan Emanuel

West University of Timisoara

This talk is based on a joint work with Zoltán SEBESTYÉN, Loránd Eötvös University of Budapest, Hungary.

Following the classical terminology of M. H. Stone [stone], two linear operators $S$ and $T$ acting between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively $\mathcal{K}$ and $\mathcal{H}$ are said to be adjoint to each other, in symbols $S\wedge T$, if $\langle Sh,k\rangle=\langle h,Tk\rangle$ for every $h\in\mathrm{dom} S$ and $k\in\mathrm{dom} T$. In our approach the range of the operator matrix $M_{S,T}=\begin{pmatrix}1_{\mathrm{dom} S} & -T\\ S & 1_{\mathrm{dom} T}\end{pmatrix}$ is playing a central role. We firstly show that $S=T^*$ and $S^*=T$ if and only if $S\wedge T$ and $\mathrm{ran}(M_{S,T})=\mathcal{H}\times\mathcal{K}$ if and only if $S, T$ are closed and densely defined, $S\wedge T, \mathrm{ran} (1+ST)=\mathcal{K}$ and $\mathrm{ran} (1+TS)=\mathcal{H}.$ We obtain, as a consequence, that for a given operator $S$ on $\mathcal{H}$ the following conditions are equivalent: $S$ is selfadjoint; $S\wedge S$ and $\mathrm{ran} M_{S, S}=\mathcal{H}$; $S$ is closed and densely defined, $S\wedge S$ and $\mathrm{ran} (1+S^2)=\mathcal{H}.$

In the case when the Hilbert space $\mathcal{H}$ is complex we revise the well-known criterion for selfadjointness of von Neumann [neu30] by dropping the assumption that the operator in discussion is a priori'' densely defined. More exactly, we show that $S$ is selfadjoint if and only if $S\wedge S, \mathrm{ran} (\lambda i+S)=\mathcal{H}$ and $\overline{\mathrm{ran} (\lambda i-S)}=\mathcal{H}$ for a certain (and also for all) $\lambda\in\mathbb{R}, \lambda\ne 0.$ Similar characterizations are obtained for skewadjoint operators (i.e., operators $S$ which verify the relation $S^*+S=0$).

Provided that $S$ and $T$ are adjoint to each other and $M_{S, T}$ has dense range, we prove, in the final part, that $T$ has dense range if and only if $S$ is closable and, if either of these statements is satisfied, the closure of $S$ equals the adjoint of $T.$ This can be seen as a generalization of the classical theorem of J. von Neumann [neu32] which characterizes closable operators. As applications we obtain new conditions which are equivalent to essential selfadjointness. To be more precise, a given linear operator $S$ on $\mathcal{H}$ is essentially selfadjoint if and only if $S$ is densely defined/closed, $S\wedge S$ and the range of $M_{S, S}$ is dense. This last condition can be replaced by $-\lambda^2\in\sigma_p(S^{*2})$ for a certain (and also for all) $\lambda\in\mathbb{R}, \lambda\ne 0.$ Also, using another approach based on a result of Arens [are61], we prove that $S$ is essentially selfadjoint if and only if $S$ is closable, $S\wedge S, \{ \mathrm{ran} S\}^\perp=\ker \bar{S}$ and $\{ h\in\mathcal{H} | \sup \{ | \langle h, h'\rangle : h'\in \mathrm{dom} S$ and $\| Sh'\|\le 1\}<\infty\}=\mathrm{ran} \bar{S}.$

[are61]
R. Arens, Operational calculus of linear relations, Pacific J. Math. 9 (1961), 9--23.

[neu30]
J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Functionaloperatoren, Math. Ann. 102 (1929-1930), 49--131.

[neu32]
J. von Neumann, Über adjungierte Funktionaloperatoren, Annals of Math. 33 (1932), 294--310.

[stone]
M. H. Stone, Linear Transformations in Hilbert Spaces and their Applications to Analysis, Amer. Math. Soc. Colloq. Publ., vol. 15, Amer. Math. Soc. 1932.

On the hyperreflexivity of power partial isometries

University of Agriculture in Krakow

The concept of reflexivity and hyperreflexivity arises from the problem of existence of a nontrivial invariant subspace for an operator on a Hilbert space. An operator is called reflexive if it has so many invariant subspaces that they determine the membership in the algebra generated by the given operator. An operator is hyperreflexive (much stronger property than reflexivity) if the usual distance from any operator to the algebra generated by the given operator can be controlled by the distance given by invariant subspaces.

A power partial isometry is an operator for which all its powers are partial isometries. Necessary and sufficient conditions for hyperreflexivity of completely non--unitary power partial isometries are given.

Joint work with K. Piwowarczyk

Dilation theory and real algebra

University of California at Santa Barbara and Nanyang Technological University

Real algebra deals with polynomial inequalities while dilation theory deals with operator inequalities and their geometric interpretation. A link between the two was established by F. Riesz, in his proof of the spectral theorem for unitary transformations, based on Riesz—Fejér factorization of non-negative trigonometric polynomials.

Most of the advances in dilation theory can be, and will be in this talk, traced to their real algebra roots. We will discuss from this perspective the celebrated Sz.-Nagy dilation theorem and Sz.-Nagy—Foias commutant lifting theorem.

Some recent advances on multivariate positivity versus joint subnormality will be mentioned, as well as an application to the construction of tight wavelet frames.

Can operator theory be used in the twin prime conjecture?

MTA Rényi Alfréd Institute and Kuwait University

By the Prime Number Theorem the average difference between consecutive primes is $p_{n+1}-p_n \sim \log p_n$. The twin prime conjecture, on the other hand, postulates that gaps between consecutive primes are 2 infinitely often (i.o. further on).

The gaps between primes can be arbitrarily small compared to average, i.e. $p_{n+1}-p_n = o(\log p_n)$ i.o. . This is the celebrated result of Goldstone, Pintz and Yildirim from 2005. It caused a real sensation echoed even in the Science and Nature magazines. The "GPY method" was summarized in a survey article by Soundararajan, who pointed out the key role of the choice of a certain polynomial $P$, underlying in the construction of the weights $\lambda_d$ in the Selberg sieve used. It is pointed out that a certain quantity $S(k)$, to be maximized for $P$, can never reach $4/k$, which is a crucial bound where even boundedness of $p_{n+1}-p_n$ -- the "Quasi Twin Prime Conjecture" -- would follow. This also showed that even the GPY method, however successful, has some theoretical limitations.

Later, Goldstone, Pintz and Yildirim achieved even $p_{n+1}-p_n = O(\log^{1/2+\varepsilon} p_n)$ with a more refined weight (and further technical advances of the method). Also, they have shown how the Elliot-Halberstam Conjecture could lead even to bounded prime gaps by means of the GPY method.

Our aim here is to solve the arising optimization problem, initiated by Soundararajan, for the choice of the polynomial $P$.

First we reformulate to a maximization (norm) problem on $L^2[0,1]$ for a self-adjoint operator $T$, the norm of which is then the maximal eigenvalue of $T$. To find eigenfunctions and eigenvalues, we derive a differential equation which can be explicitly solved. We find $S(k)=4/(k+ck^{1/3})$, achieved by the $k-1^{\textrm{st}}$ integral of $x^{1-k/2}J_{k-2}(\alpha_{1}\sqrt{x})$, where $\alpha_1\sim c k^{1/3}$ is the first positive root of the $k-2^{\rm nd}$ Bessel function $J_{k-2}$.

As this naturally gives rise to a number of technical problems in the application of the GPY method, we also construct a better manageable and approximately optimal polynomial $P$, furnishing an approximately optimal extremal quantity $4/(k+Ck^{1/3})$ with some other constant $C$.

In the forthcoming paper of J. Pintz it is indeed shown how this quasi-optimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits. The result is that we even have $p_{n+1}-p_n = O(\log^{3/7+\varepsilon} p_n)$ i.o., which is the theoretical limit of the GPY method.

Submodules and quotient modules of the Hardy module over polydisc

Indian Statistical Institute

In this talk we will discuss a complete classification of the doubly commuting quotient modules and submodules of the Hardy module over polydisc. We also describe the issue of essential doubly commutativity and the rigidity phenomenon for a class of submodules of the Hardy module over polydisc.

Compatible pairs of commuting isometries

Jagiellonian University

In the paper the pair of commuting isometries is considered. The decomposition of the pair onto compatible and completely non compatible part is given. The structure and decompsition of wanderin subspaces for compatible part is investigated. Examples and model for the compatible part are shown.

This is a joint work with Zbigniew Burdak and Marek Kosiek.

On strongly continuous one-parameter groups of holomorphic unit ball automorphisms

University of Szeged

We prove a structure theorem for strongly continuous one-parameter groups formed by surjective isometries of the space of bounded $N$-linear functionals over complex Hilbert spaces. Though the result is a natural analogue to Stone's classical theorem covering the case $N=1$, some crucial arguments of the proof go back to probability theory. As a consequence, we classify the strongly continuous one-parameter automorphism groups of all infinite-dimensional Cartan factors of Jordan theory. We reduce the investigation of the strongly continuous one-parameter groups by non-linear holomorphic automorphism of the unit ball in ${\mathcal{L}}(H,K)$ to the study of some retarded ordinary differential equations.

Ergodic results for Cesàro and Abel bounded operators

Suciu, Laurian

"Lucian Blaga" University of Sibiu

Co-Authors : Michael Lin, David Shoikhet, J. Zemánek.

New growth conditions on the Cesàro means of higher order are investigated for Banach space operators with peripheral spectrum reduced to {1}. These topics are related to Gelfand-Hille and Esterle-Katznelson-Tzafriri type theorems.

We also study uniform ergodicity for not necessarily power-bounded operators, and relate it to the uniform convergence of the Abel averages.

Finally, Poisson's equation for mean ergodic operators is presented. As an application in reflexive spaces, we supplement the Fonf-Lin-Wojtaszczyk characterization of reflexivity of Banach spaces with a basis.

The algebraic associate of Szőkefalvi-Nagy's influential dilation theorem

Jagiellonian University

The Sz.-Nagy dilation theorem (1953) is one of the most celebrated theorems in Operator Theory. The other, coming from 1955, seems to be in its shadow, if not forgotten. My intension is to revive it showing the links to another distinguished theorems of those times as well as drawing up its far-reaching consequences.

An extremal problem for characteristic functions

Institute of Mathematics Simion Stoilow of the Romanian Academy

Let $E$ be a subset of the unit circle. We will discuss the problem of evaluating the quantities $\Lambda_n(E)=\sup \left\{\left|\int_E F(\zeta)\bar\zeta^n\, d\zeta\right| : F\in H^1,\ \|F\|_1\le 1\right\}.$ This is joint work with I. Chalendar, S. Garcia, and W. Ross.

„Large” weak orbits of $C_0$-semigroups

Tomilov, Yuri

IM PAN, Nicolaus Copernicus University

It is a well-known heuristic principle that the Fourier transform is as large as it is allowed to be by very basic constraints. We will discuss several operator-theoretical generalizations of this principle. Under natural spectral assumptions, we will show that if all weak orbits of a Hilbert space $C_0$-semigroup decay to zero then the decay is arbitrarily slow. Similar results and their applications to the theory of Fourier transforms will be discussed as well.

This is joint work with V. Müller.

A Wolff type Theorem for Ideals in the Mulplier Algebra on a Weighted Dirichlet Space

University of Alabama

In an effort to classify ideal membership in $H^{\infty}(\mathbb{D})$, T. Wolff [garnett] showed that if \begin{align} \{f_j\}_{j=1}^n \subset H^{\infty}(\mathbb{D}), H \in H^{\infty}(\mathbb{D}) \quad \text{and}\notag\\ |H(z)| \le \left( \sum_{j=1}^n \, |f_j(z)|^2\right)^{\frac 1{2}} \;\; \text{for all } \, z \in \mathbb{D}, \notag \end{align} then $H^3 \in \mathcal{I} ( \{ f_j\}_{j=1}^n),$ the ideal generated by $\{f_j\}_{j=1}^n$ in $H^{\infty}(\mathbb{D})$.

Note that the "3" above cannot be replaced, in general, by "1" [garnett], or even "2" [treil]. We establish the analogous result for the algebra of multipliers on a weighted Dirichlet space. Our proof breaks into 2 parts. One part uses the fact that the reproducing kernel of a weighted Dirichlet space is a complete Nevanlinna-Pick kernel. This enables us to use an important result from Sz.Nagy-Foias model theory.[sznf]

[garnett]
Garnett,J.B., Bounded Analytic Functions, Academic Press, 1981.

[sznf]
Sz.Nagy,B. ; Foias,C. , Harmonic Analysis of Operators on Hilbert Space, North Holland, 1970

[treil]
Treil,S.R., Estimates in the corona theorem and ideals in $H^{\infty}$; A problem of T. Wolff, Academic Press, J. Anal. Math. 87 (2002), 481-495. 1981.

Aluthge transform for unbounded operators

Jagiellonian University

For a bounded operator $T$ with polar decomposition $T=U|T|$, the Aluthge transform of $T$ is the operator $\widetilde{T}=|T|^{\frac12}U|T|^{\frac12}$. I try to generalize this notion to densely defined closed operators. I show that the Aluthge transform of such an operator need not be densely defined or closable.

Block numerical range (BNR) of analytic block operator matrix functions

University of Bern

In this talk the quadratic and more generally block numerical range of analytic block operator matrix functions is presented. The main results include the spectral inclusion property and resolvent estimates; they generalize corresponding results for the numerical range of operator functions.

Singular neutral functional differential equations and product spaces

University of Texas at Dallas

Hereditary systems including various classes of retarded and neutral equations have been studied using well-posed state space formulations. In that abstract setting product spaces have played a significant role in the development and analysis of finite dimensional approximations corresponding to these infinite dimensional problems. A key issue there is the availability of a dissipativity estimate for the infinitesimal generator of the associated semigroup. For singular neutral equations (i.e., equations with non-atomic difference operator) well-posedness was shown on certain product spaces, but no dissipativity estimates were obtained. In order to overcome this shortcoming (which is serious when it comes to numerics) weighted Lebesgue spaces have been suggested as state spaces for a class of singular neutral equations. In this talk we describe the procedure which provides, via extension starting from the weighted Lebesgue space, a state space with product space structure for singular neutral equations. In particular, we show that this state space is the extrapolation space corresponding to the solution semigroup on the weighted Lebesgue space.

Dual functional models of linear operators of Nagy–Foiaş type

We give a review of the construction of a linearly similar model, suggested a few years ago by the speaker in [Yaku-A-A, Yaku-JOT], and their applications to sectorial operators [GMY]. This construction generalizes the $C_{00}$ case of the Nagy—Foiaş construction and in some sense, has more flexibility. The model can be constructed not only in a disc or a semiplane, as in the original Nagy—Foiaş setting, but rather in a wide class of domains in the complex plane. Even if the domain is fixed, the model is far from unique. Nor is unique the (generalized) characteristic function, which determines the model. The main ingredient to choose are two auxiliary operators, which are the analogues of the Nagy-Foiaş defect operators.

The models of an operator and of its adjoint are given in two different functional spaces, and there is a natural Cauchy duality between these spaces. In general, the model operator we obtain is only similar and not unitarily equivalent to the original one.

Examples of concrete applications include non-dissipative unbounded perturbations of unbounded selfadjoint operators, generators of semigroups related to delay equations and all generators of $C_0$ groups.

Main attention will be paid to generators of analytic semigroups. Our main result is that a sectorial operator admits an $H^\infty$-functional calculus if and only if it has a functional model of Nagy-Foiaş type. We give a simple concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any Hilbert space sectorial operator by passing to a different norm (the McIntosh square function norm). We show that this norm is close to the original one, in the sense that there is only a logarithmic gap between them.

In this case, the dual functional models have the sense of an observation model and a control model and can be defined by basic linear control theory.

The talk is based on the joint work [GMY] with José Galé and Pedro Miana from the University of Zaragoza.

[Yaku-A-A]
D. V. Yakubovich, A linearly similar Sz. Nagy-Foiaş model in a domain. Algebra i Analiz 15 (2003), no. 2, 190-237; English transl.: St. Petersburg Math. J. 15 (2004), no. 2, 289-321.

[Yaku-JOT]
D. V. Yakubovich, Nagy-Foiaş type functional models of nondissipative operators in parabolic domains. J. Oper. Theory 60 (2008), no. 1, 3-28.

[GMY]
J. E. Galé, P. J. Miana, D. V. Yakubovich, $H^\infty$-functional calculus and models of Nagy-Foiaş type for sectorial operators. Math. Ann. 351 (2011), no. 3, 733-760.

Nagy–Foias model theory in the Hardy space over the bidisk

State University of New York at Albany

Most part of Nagy-Foias operator model theory was laid out in the vector-valued Hardy space $H^2(E)=H^2({\mathbb D})\otimes E$, where ${\mathbb D}$ is the unit disk and $E$ is a separable Hilbert space. If $E=H^2({\mathbb D})$, then $H^2(E)$ is the Hardy space over the bidisk $H^2({\mathbb D}^2)$ which clearly has more structure. Key elements in the model theory, such as defect operator and characteristic operator function, now have a new look. This translation brings new methods into the study of invariant subspaces in $H^2({\mathbb D}^2)$. As a particular result, the two pathological invariant susbpaces that Rudin constructed over 50 years ago now turn out to be pretty nice.

Hyperreflexivity constants of some spaces of matrices

Zajac, Michal

Faculty of Electrical Engineering and Information Technology STU

Based on a joint work with Janko Bračič and Viktória Rozborová [B-R-Z].

Let $X$ be a complex Banach space and $B(X)$ denote the set of all bounded linear operators on $X$. For a subspace $\mathcal{S}\subset B(X)$ and any $A\in B(X)$ let $$\mathrm{dist}(A,\mathcal{S})=\inf_{S\in\mathcal{S}}\|A-S\|=\inf_{S\in\mathcal{S}}\sup_{\|x\|=1}\|(A-S)x\|$$ be the usual distance and $$\alpha(A,\mathcal{S})=\sup_{\|x\|=1}\inf_{S\in\mathcal{S}}\|(A-S)x\|$$ the Arveson distance of $A$ to $\mathcal{S}$. It is obvious that $\alpha(A,\mathcal{S})\leq\mathrm{dist}(A,\mathcal{S})$. If $\mathrm{Ref}\mathcal{S}=\{A\in B(X): \alpha(A,\mathcal{S})=0\}=\mathcal{S}$, then the subspace $\mathcal{S}$ is reflexive. If there exists a positive constant $c>0$ such that $$\label{hyperrefl} (1) \qquad\qquad \mathrm{dist}(A,\mathcal{S})\le c \alpha(A,\mathcal{S})\quad \text{ for all }A\in B(X)\,,$$ then $\mathcal{S}$ is said to be hyperreflexive and the smallest $c\ge1$ for which (1) holds, i.e. $$\label{hyerref-c} (2) \qquad\qquad \kappa(\mathcal{S})=\sup_{A\in B(X)\setminus\mathrm{Ref}\mathcal{S}}\frac {\mathrm{dist}(A,\mathcal{S})}{\alpha(A,\mathcal{S})}$$ is the constant of hyperreflexivity of $\mathcal{S}$.

It is known [Mags] that on Hilbert spaces all 1-dimensional spaces of operators are hyperreflexive with $\kappa(\mathcal{S})=1$. This is not true in Banach space. We give an example of 1-dimensional subspace $\mathcal{S}$ of $2\times 2$ matrices considered as operators on space $\mathbb{C}^{2\times1}$ endowed with $\ell_1$ norm $\|(x_1,x_2)^{\top}\|_1=|x_1|+|x_2|$ for which $\kappa(\mathcal{S})=\sqrt2$. Our main result is Theorem 6 in [B-R-Z]:

Theorem 1. Let $I\in \mathbb{C}^{2\times2}$ be the identity matrix and $\mathcal{S}=\{\lambda I:\lambda\in \mathbb{C}\}$. Then $\kappa(\mathcal{S})=\sqrt2$.

To prove Theorem 1 we formulate and use some interesting geometric results. First, we show that for $A_0=\left(\begin{smallmatrix} 2&1\\1&0\end{smallmatrix}\right)$ $$\mathrm{dist}(A_0,\mathcal{S})=2\,,\qquad \alpha(A_0,\mathcal{S})=\sqrt2$$ which, by (2), implies that $\kappa(\mathcal{S})\ge\sqrt2$. The reverse inequality is then proved by showing that for all $A\in \mathbb{C}^{2\times2}\setminus\mathcal{S}$ $$\frac {\mathrm{dist}(A,\mathcal{S})}{\alpha(A,\mathcal{S})}\le\sqrt2\,.$$

Supported by grant VEGA 1/0426/12 of the Ministry of Education of Slovak Republic

[B-R-Z]
J. Bračič, V. Rozborová, M. Zajac, Hyperreflexivity constants of some spaces of matrices, Linear Algebra Appl. (2013) http://dx.doi.org/10.1016/j.laa.2013.04.017

[Mag]
B. Magajna, On the distance to finite-dimensional subspaces in operator algebras, J. London Math. Soc. 47 (1993), 516-532.

On Cesàro ergodicity of linear operators

Zemánek, Jaroslav

Institute of Mathematics of the Polish Academy of Sciences

In this joint work with Laurian Suciu, we present an operator that is not power-bounded, but whose every power is Cesàro ergodic. This is obtained within our improvements on the 1939 Lorch theorem.

Minimal extrapolation for Fourier transforms

University of Rome "Tor Vergata"

Let $G$ be an abelian locally compact group and $\widehat{G}$ its dual group. If $F\subset \widehat{G}$ is a closed set and the function $\varphi : F\longrightarrow \mathbb{C}$ can be extended to the Fourier transform $\widehat{g}$ of some $g\in L^1(G)$ then the following "minimal extrapolation" problem can be considered:
• compute $\alpha =\inf\big\{ \| g\|_{1}\, ; g\in L^1(G)\, , \widehat{g}\;\!\big|_F =\varphi\big\}\, ;$
• decide if the above infimum is a minimum, and if yes,
• describe all $f\in L^1(G)$ with $\widehat{f}\;\!\big|_F =\varphi$ for which $\| f\|_{1}=\alpha\, .$

The above minimal extrapolation problem was first considered by Arne Beurling [ab] and Béla Szőkefalvi-Nagy ([szns], [szn1], [szn2]) in the second half of the decade 1930 - 1940 and for a long time the work of Sz.-Nagy yielded the only method to solve explicitely this kind of problems. In this talk we intend to discuss their results, subsequent developments and some applications.

[ab]
Beurling, Arne, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionnelle. 9-ième Congrès des mathématiciens scandinaves, Helsingfors, 1938.

[szns]
Sz.-Nagy, Béla; Strausz-Sólyi, Arthur, H. Bohr egy tételéről. Mat. Természettudományi Értesítő 57 (1938), 121-133.

[szn1]
Sz.-Nagy, Béla, Über gewisse Extremalfragen bei transformierten trigono- metrischen Entwicklungen, I. Periodischer Fall. Ber. Math. Phys. Kl. Sächs. Akad. Wiss. 50 (1938), 103-134.

[szn2]
Sz.-Nagy, Béla, Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen, II. Nichtperiodischer Fall. Ber. Math. Phys. Kl. Sächs. Akad. Wiss. 51 (1939), 3-24.