IWOTA 2019

International Workshop on

Operator Theory and its Applications

IWOTA 2019

International Workshop on

Operator Theory and its Applications

July 22-26, 2019

Instituto Superior Técnico

Lisbon, Portugal

Instituto Superior Técnico

Lisbon, Portugal

- Vladimir Bolotnikov

College of William & Mary, Williamsburg, Virginia, USA ## On a characterization of shift operator tuples on certain weighted Bergman-Fock spaces

As was discovered by A. Olofsson and O. Giselsson, the shift operator $S_n$ on the standard weighted Bergman space $A_n$ satisfies the identity $$(S_n^*S_n)^{-1}=\sum_{k=0}^{n-1}(-1)^k\left(\begin{array}{c}n \\ k+1\end{array}\right)S_n^kS_n^{*k},$$ which, under the extra pureness assumption, characterizes the Bergman shift up to unitary similarity.

We will discuss an extension of the Olofsson-Gilellson identity to the non-commutative setting of weighted Bergman Fock spaces and a related characterization of (right) shift operator tuples on these spaces.

- Raúl E. Curto

University of Iowa ## Quasinormality of powers of commuting pairs of bounded operators

We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal $2$-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator $T$ whose square $T^2$ is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal $n$-tuples.

Fourth, we show that if a $2$-variable weighted shift $W_{(\alpha,\beta)}$ and its powers $W_{(\alpha,\beta)}^{(2,1)}$ and $W_{(\alpha,\beta)}^{(1,2)}$ are all spherically quasinormal, then $W_{(\alpha,\beta)}$ may not necessarily be jointly quasinormal. Moreover, it is possible for both$W_{(\alpha,\beta)}^{(2,1)}$and $W_{(\alpha,\beta)}^{(1,2)}$ to be spherically quasinormal without $W_{(\alpha,\beta)}$ being spherically quasinormal. Finally, we prove that, for $2$-variable weighted shifts, the common fixed points of the toral and spherical Aluthge transforms are jointly quasinormal.

The talk is based on joint work with S. H. Lee and J. Yoon.

- Michael Dritschel

Newcastle University ## The rational dilation problem on distinguished varieties

It is well known that the Sz.-Nagy dilation theorem and the von Neumann inequality on the unit disk are equivalent, and that this in turn implies a matricial form of the von Neumann inequality holds for matrix valued functions of all dimensions. Ando’s theorem implies a similar result for the bidisk and pairs of commuting operators, and Arveson later proved a general principle that a tuple of commuting operators with spectrum in a compact set dilates to a commuting tuple of operators with spectrum on the boundary if and only if a matricial von Neumann inequality holds on the set. Since the complement of the set may have several components, one uses algebras of matrix valued rational functions with poles off of the set, and the dilation is referred to as a rational dilation. The rational dilation problem asks: Does a scalar von~Neumann inequality suffices for rational dilation to hold? The speaker, with McCullough and Jury showed that, somewhat surprisingly, the answer is

*no*on the Neil parabola, a distinguished variety in the bidisk. We discuss here further work with Batzorig Undrakh which illustrates the ubiquity of the failure of rational dilation on distinguished varieties.- Jörg Eschmeier

Saarland University ## Bergman spaces and $m$-hypercontractions

We give a survey on some recent results from multivariable operator theory on the unit ball in $\mathbb C^n$ including transfer function realizations of Bergman-inner functions, characterizations of m-shifts by Wold-type decompositions and characterizations of Toeplitz operators with pluriharmonic symbol in terms of higher order Brown-Halmos conditions. A part of the results is based on joint work with Sebastian Langendoerfer.

- George Exner

Bucknell University ## Subnormality and infinite divisibility of weighted shifts

A subnormal weighted shift $W_\alpha$ with weight sequence $\alpha = (\alpha_n)_{n=0}^\infty$ is infinitely divisible if the weight sequence $\alpha^{(p)} = (\alpha_n^p)_{n=0}^\infty$ yields a subnormal shift for each $p > 0$. We exhibit several new necessary and sufficient conditions for infinite divisibility, produce new examples of subnormal and infinitely divisible shifts, consider Schur flows of shifts, and explore connections with completely hyperexpansive shifts. As well we provide new conditions for subnormality and exhibit an enlightening connection between the $k$-hyponormality ($k=1, 2, \ldots$) and $n$-contractivity ($n=1, 2, \ldots$) conditions.

(Joint work with Raul Curto and Chafiq Benhida)

- Anatoly Golberg

Holon Institute of Technology, Israel ## Mappings with integrally controlled moduli

In the talk we discuss the differential properties of multidimensional homeomorphic/open discrete mappings. Such mappings (multivariable operators) essentially generalize the well-known customarily investigated classes of mappings as quasiregular, quasiisometric, Lipschitzian, etc. But in contast to these known classes, the definion of our mapping class does not involve any analytic restrictions. We also illustrate the regularity properties by several examples and present a collection of open related problems.

The talk is based on joint works with R. Salimov (Institute of Mathematics, Kyiv, Ukraine).

- Estiaan Klem

North-West University and Vrije Universiteit Amsterdam ## Cauchy-Riemann equations for free noncommutative functions

In classical complex analysis analyticity of a complex function $f$ is equivalent to differentiability of its real and imaginary parts $u$ and $v$, respectively, together with the Cauchy-Riemann equations for the partial derivatives of $u$ and $v$. We extend this result to the context of free noncommutative functions on tuples of matrices of arbitrary size. In this context, the real and imaginary parts become so called real noncommutative functions, as appeared recently in the context of Löwner's theorem in several noncommutative variables. Additionally, as part of our investigation of real noncommutative functions, we show that real noncommutative functions are in fact noncommutative functions.

- Zinaida Lykova

Newcastle University ## The construction of rational tetra-inner functions

A tetra-inner function is a holomorphic map $x=(x_1,x_2,x_3)$ from the unit disc $\mathbb{D}$ to the closed tetrablock $\overline{\mathcal{E}}$, whose boundary values at almost all points of the unit circle $\mathbb{T}$ belong to the distinguished boundary $b\overline{\mathcal{E}}$ of $\overline{\mathcal{E}}$. Here \[ \overline{\mathcal{E}}=\{x∈\mathbb{C}^3:1−x_1z−x_2w+x_3zw≠0 \quad\text{whenever}\quad |z|<1,|w|<1 \}. \]

There is a natural notion of degree of a rational tetra-inner function $x$; it is simply the topological degree of the continuous map $x_\mathbb{T}$ from $\mathbb{T}$ to $\overline{\mathcal{E}}$. In this talk we give a prescription for the construction of a general rational tetra-inner function of degree $n$. The prescription makes use of a known solution of an interpolation problem for finite Blaschke products of given degree in terms of a Pick matrix formed from the interpolation data. It turns out that a natural choice of data for the construction of a rational tetra-inner function $x=(x_1,x_2,x_3)$ consists of the points in $\overline{\mathbb{D}}$ for which $x_1x_2−x_3=0$ and the values of $x$ at these points.

The talk is based on joint work with my PhD student Hadi Alshammari.

- Leila Nasiri

Lorestan University, Mathematics, Iran. ## The reverses of Young type inequalities

In this note, we give the further reverses of the Young inequalities for non-negative real scalars.

Making use of them, some matrix inequalities for Hilbert-Schmidt norm and trace norm are deduced.

- Meredith Sargent

University of Arkansas ## Escaping Non-Tangential Approach

A classical Julia-Carathéodory theorem states that if there is a sequence tending to $\tau$ in the boundary of a domain $D$ along which the Julia quotient is bounded, then the function $\phi$ can be extended to $\tau$ such that $\phi$ is nontangentially continuous and differentiable at $\tau$ and $\phi(\tau)$ is in the boundary of $\Omega$.

We develop a theory in the case of Pick functions where we consider sequences that approach the boundary in a controlled tangential way, yielding necessary and sufficient conditions for higher order regularity. In this talk, we discuss some of the technical details involved, including amortization of the Julia Quotient, $\gamma$-regularity, and $\gamma$-auguries.

- Dominik Schillo

Universität des Saarlandes ## A Beurling Theorem for Spaces with a CNP Factor

In this talk, we present a generalization of the Beurling–Lax–Halmos-type theorem of McCullough and Trent for reproducing kernel Hilbert spaces whose kernel has a complete Nevanlinna–Pick factor. We also record factorization results for pairs of nested invariant subspaces.

This is joint work with Raphael Clouatre and Michael Hartz.

- Edward John Timko

University of Manitoba ## Spectral Properties of Constrained Row Contraction

In this talk we discuss some results on the spectral properties of constrained absolutely continuous commuting row contractions in relation to the constraining ideal. This is joint work with Raphael Clouatre.

- Dmitry Yakubovich

Universidad Autónoma de Madrid, Departamento de Matemáticas ## Perturbations of normal operators, their Sobolev space models and questions of geometric measure theory

Let $N$ be a bounded normal operator on a separable Hilbert space and let $\mu$ stand for its scalar spectral measure. The spectral nature of the perturbation $T=N+K$, where $K$ is a sufficiently "smooth", finite rank compact operator, will be discussed. We are interested in the existence of invariant subspaces, decomposability and other questions.

We introduce the perturbation matrix-valued function of $T$, defined in the whole complex plane, except for a certain thin set, and explain its role. Our main tool is the construction of a certain quotient model of $T$, defined in terms of certain spaces of Cauchy integrals. We discuss the dependence of the answers on geometric properties of $\mu$.

The case when $\mu$ is absolutely continuous with respect to the area measure has been considered in [1] in 1993; this is the case of $\mu$ of "dimension" $2$. A quotient model for $T$, constructed in terms of certain vector-valued Sobolev classes of functions, was established in this work. (In that paper, infinite rank perturbations were also dealt with.) The case of a discrete measure $\mu$ (that is, of a diagonalizable operator $N$) has been studied more recently in a series of papers by C. Foias and his coauthors (see [2]). This can be seen as the case of zero dimension. The case when $\mu$ is of "dimension" $1$ (that is, when $\mu$ behaves like the arc length measure) seems to be the most difficult one, whereas many cases of fractional dimension are easier.

This is a joint work in progress with Mihai Putinar.

### References

- D. Yakubovich, Spectral properties of smooth perturbations of normal operators with planar Lebesgue spectrum, Indiana Univ. Math. J. 42, no. 1 (1993), 55-83.
- C. Foias, I. Jung, E. Ko, C. Pearcy, Spectral decomposability of rank-one perturbations of normal operators, J. Math. Anal. Appl. 375 (2011), 602-609.