Joseph A Ball
Pairs of commuting contractions: Ando lifts and functional models
decomposition for an isometric Hilbert-space operator leads to a
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
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
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
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
function for the single product contraction operator $T$, (ii) the BCL
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
unitary invariant for $(T_1, T_2)$ but with the characteristic function
$T = T_1 T_2$
dropped). There is a special class of Ando
lifts for which it can be proved that the additional operator equations
satisfied, leading to a new functional-model proof of the Ando dilation
hold for commuting contractive $d$-tuples. The fact that the Ando
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.
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.
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.
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.
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.
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