Speakers, titles, abstracts

Joseph A Ball

Pairs of commuting contractions: Ando lifts and functional models

The Wold decomposition for an isometric Hilbert-space operator leads to a functional model for such an operator: the shift operator on a vectorial Hardy space direct sum with a unitary operator for which there is a good spectral theory (complete unitary invariants consisting of the scalar spectral measure together with multiplicity function on the unit circle). Sz.-Nagy and Foias (late 1960s) showed how to construct a functional model for a completely non-unitary contraction operator $T$ by understanding how to embed such a functional-model space into the Wold-decomposition functional-model for the single product isometry $V=V_1 V_2$.

This story extends to the setting of commuting pairs of isometries and commuting pairs of contractions as follows. Berger-Coburn-Lebow (1978) found a functional model for a commuting pair of isometries $(V_1, V_2)$ by using the Wold-decomposition functional-model for the single product isometry $V = V_1 V_2$ as the ambient functional-model space. They also found a complete unitary invariant (a ``BCL tuple” consisting of a Hilbert coefficient space together with a unitary and projection operator on this space) from which one can construct the functional model. In the case where the product isometry is pure, $V$ is multiplication by the coordinate function $z$ while $V_1$ and $V_2$ are equal to multiplication by certain operator pencils with coefficients coming from the BCL tuple.

In this talk we discuss the next step: finding a functional model for a commutative pair of contraction operators $(T_1, T_2)$ by seeing how to embed such a space into the BCL functional model for its minimal Ando isometric lift $(V_1, V_2)$. This amounts to finding the models for the commutative operator pair $(T_1, T_2)$ as acting on the Sz.-Nagy-Foias functional-model space for the single product operator $T = T_1 T_2$. Our approach to the Sz.-Nagy-Foias model was inspired by the approach of Douglas (1968). A complete invariant consists of (i) the Sz.-Nagy-Foias characteristic function for the single product contraction operator $T$, (ii) the BCL tuple for the Ando lift $(V_1, V_2)$, and (iii) an embedding operator $\Lambda$ of the defect space of $T^*$ into the defect space of $V^* = V_1^* V_2^*$ which again is subject to some additional operator equations.

Part of this construction is a functional model for any Ando lift $(V_1, V_2)$ for the commutative contractive pair $(T_1, T_2)$. Ando lifts are classified by a collection of spaces and operators which we call Ando tuples (the complete unitary invariant for $(T_1, T_2)$ but with the characteristic function $\Theta_T$ for

$T = T_1 T_2$ dropped). There is a special class of Ando lifts for which it can be proved that the additional operator equations are automatically satisfied, leading to a new functional-model proof of the Ando dilation theory itself.

Parallel results hold for commuting contractive $d$-tuples. The fact that the Ando dilation theorem does not extend in general to commutative contractive $d$-tuples once $d >2$ is explained by the fact that the auxiliary supplementary operator equations fail to hold in general.

This is joint work with Haripada Sau of the Indian Institute of Technology Guwahati.

Alain Connes

Prolate wave operator and Zeta

I will explain in my talk the recent work with Henri Moscovici on the link, in the ultraviolet regime, between the spectrum of a natural self-adjoint extension of the classical prolate wave operator and the squares of zeros of the Riemann zeta function.

Fritz Gesztesy

Continuity properties of the spectral shift function for massless Dirac operators and an application to the Witten index

We report on recent results regarding the limiting absorption principle for multi-dimensional, massless Dirac-type operators (implying absence of singularly continuous spectrum) and continuity properties of the associated spectral shift function.

We will motivate our interest in this circle of ideas by briefly describing the connection to the notion of the Witten index for a certain class of non-Fredholm operators.

This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and D. Zanin.

Javad Mashreghi

Dilation Theory, a one-century-old story

This is a friendly recount of dilation theory in the complex plane. Starting with the pioneering result of F. Riesz (1923) in the Hardy space $H^p$, we provide a wide range of theorems which are uniformly spread over a century to tackle the problems in different function spaces such as the Bergman space, super-harmonically weighted Dirichlet spaces, the Bloch space, model spaces, de Branges-Rovnyak spaces, etc. We finish with some recent results and open questions, emerged in the last decade, to highlight the need for novel tools in dealing with such problems.

Mihai Putinar

The legacy of Carleman's doctoral dissertation

Carleman's name is linked to primary results and techniques of XX-th century mathematical analysis.

Less known is his doctoral dissertation "On the Neumann-Poincare problem on a domain with corners" defended when he was 24 years old. This highly original work, barely cited or read in detail, marks a notable qualitative leap in spectral analysis. Motivated by the ever challenging Dirichlet problem on non-smooth boundaries, Carleman interlaces the layer potentials approach, advocated by Carl Neumann and greatly advanced by Hilbert, with a variational principle formulated by Poincare. The result is a refined spectral analysis of a concrete problem, a true tour de force synthesizing in surprising unity the circulating mathematical techniques of the turn of the century. Many ingredients in Carleman's dissertation would later shape his renown discoveries. The lecture will offer an overview of Carleman's thesis and the evolution of his groundbreaking methods.

If time allows, I will comment, with some brief examples, on the current explosion of interest in the applied facets of spectral analysis of the same integral operator, the one which intrigued Carelman as well as generations of scientists before and after him.

Dan-Virgil Voiculescu

Quasicentral modulus as a noncommutative nonlinear condenser capacity

The quasicentral modulus is key in many questions about normed ideal perturbations of n-tuples of operators. I have extended the definition to that of a condenser quasicentral modulus and I will point out a noncommutative analogy with condenser capacity in nonlinear potential theory.