Operator Theory, Complex Analysis and Applications Seminar

Past sessions

Invitation to weighted shifts on directed trees

Weighted shifts on directed trees form an important class of operators introduced recently in [6]. This class is a natural and substantial generalization of the class of classical (unilateral or bilateral) weighted shifts on $\ell^2$ spaces. It is also related to a class of composition operators in $L^2$-spaces.

Weighted shifts on directed trees have proven to have very interesting features (see [2, 3, 5, 6, 7]). The underlying relatively simple graph structure gives a rise to a subtle and complex structure of the operators, which turn out to have properties not known before in other classes of operators, and makes them ideal for testing hypothesises and constructing examples. We will outline recent results concerning these operators with main emphasis on on subnormality and reflexivity.

The talk is based on a joint work with Z.J. Jabloński, I.B. Jung and J. Stochel, and M. Ptak.

References

1. P. Budzyński, P. Dymek, Z. J. Jabłoski, J. Stochel, Subnormal weighted shifts on directed trees and composition operators in $L^2$-spaces with non-densely defined powers, Abstract Appl. Anal. (2014), Article ID 791817, 6 pages.
2. P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Subnormality of unbounded weighted shifts on directed trees, J. Math. Anal. Appl. 394 (2012), 819-834.
3. P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Subnormality of unbounded weighted shifts on directed trees. II, J. Math. Anal. Appl. 398 (2013), 600-608.
4. P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Unbounded subnormal composition operators in $L^2$-spaces, preprint, http://arxiv.org/abs/1310.3542.
5. P. Budzyński, Z. J. Jabłoski, I. B. Jung, J. Stochel, Subnormal weighted shifts on directed trees whose nth powers have trivial domain, preprint, http://arxiv.org/abs/1409.8022.
6. Z. J. Jabłoński, I. B. Jung, J. Stochel, Weighted shifts on directed trees, Mem. Amer. Math. Soc 216 (2012), no. 1017.
7. Z. J. Jabłoński, I. B. Jung, J. Stochel, A non-hyponormal operator generating Stieltjes moment sequences, J. Funct. Anal. 262 (2012), 3946-3980.

Poisson integrals on Riemannian Symmetric Spaces

In this talk we shall give characterizations of the $L^{p}$-range of the Poisson transform $P_{\lambda}$ associated to Riemannian Symmetric Spaces. We will focus on the rank one symmetric space case, and show that for $\lambda$ real, the Poisson transform is a bijection from the space of $L^{2}$ functions on the boundary (respectively $L^{p}$) onto a subspace of eigenfunctions of the Laplacian satisfying certain $L^{2}$-type norms (respectively Hardy-type norms).

Spectral analysis of Jacobi operators generated by Markov Birth and Death Processes

Some particular examples of Jacobi Operators (tridiagonal matrices) with growing entries related to the Markov processes will be considered.

Using a Levinson's type theorems approach we plan to determine the spectral structure of the corresponding operators.

No preliminary knowledge of Jacobi Matrices or Orthogonal Polynomials to be required.

Characterising Higher-Rank Graph C*-Algebras

There is an elegant theory for graph C*-algebras that allows one to determine structural properties of the C*-algebra from the underlying directed graph. By coupling this with C*-algebra classification results one can characterise many graph C*-algebras as falling into various known classes of nuclear classifiable C*-algebras.  Whereas much of the structural theory carries over, the C*-algebras associated to higher-rank analogues of directed graphs are much less well-understood. I will recall the standard tools that are available to study higher-rank graph C*-algebras and discuss how recent developments in Elliot's classification programme could be used to help characterise higher-rank graph C*-algebras.

Sampling, interpolation and Riesz bases in the small Fock spaces.

We give a complete description of Riesz bases and characterize interpolation and sampling in terms of densities.

This is joint work with A. Baranov, A. Dumont and A. Hartmann.

Integral operators and elliptic equations in variable exponent Lebesgue spaces.

We study mapping properties of variable order Riesz and Wolff potentials within the framework of variable exponent Lebesgue spaces. As an application, optimal integrability results for solutions to the $$p(.)$$-Laplace equation are given in the scale of (weak) Lebesgue spaces.

Pseudospectra in non-Hermitian quantum mechanics

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-self-adjoint operators. We relate pseudospectral properties to quasi-self-adjointness, similarity to self-adjoint operators and basis properties of eigenfunctions. Applying microlocal techniques for the location of the pseudospectrum of semiclassical operators to models familiar from recent physical literature, unexpected wild properties of the operators are revealed. This is joint work with Petr Siegl, Milos Tater and Joe Viola.

Riemann-Hilbert problems, Toeplitz operators and $$Q$$-classes

We generalize the notion of $Q$-classes ${C}_{{Q}_{1},{Q}_{2}}$, which was introduced in the context of Wiener-Hopf factorization, by considering very general $2×2$ matrix functions ${Q}_{1}$, ${Q}_{2}$. This allows us to use a mainly algebraic approach to obtain several equivalent representations for each class, to study the intersections of $Q$-classes and to explore their close connection with certain non-linear scalar equations. The results are applied to various factorization problems and to the study of Toeplitz operators with symbol in a $Q$-class.

Optimal bounds for analytic projections

We discuss some recent advances related to size estimates of analytic projections and the possible uses for such estimates in applications. The spaces considered include Hardy, Bergman, Bloch, Besov and Segal-Bargmann spaces. We study in detail the case of Bergman projection onto the maximal and minimal Möbius invariant spaces.

The Hua operators on homogeneous line bundles over bounded symmetric domains of tube type

Let $𝒟=G/K$ be a bounded symmetric domain of tube type. We show that the image of the Poisson transform on the degenerate principal series representation of $G$ attached to the Shilov boundary of $𝒟$ is characterized by a $K$- covariant differential operator on a homogeneous line bundle over $𝒟$. As a consequence of our result we get the eigenvalues of the Casimir operator for Poisson transforms on homogeneous line bundles over $G/K$. This extends a result of Imemura and all on symmetric domains of classical type to all symmetric domains. Also we compute a class of Hua type integrals generalizing an earlier result of Faraut and Koranyi.

Spectral analysis of Jacobi Matrices and asymptotic properties of orthogonal polynomials

We review basic features of the spectral theory of Hermitian Jacobi operators. The analysis is based on asymptotic properties of the related orthogonal polynomials at infinity for fixed spectral parameter. We discuss various examples of bounded and unbounded Jacobi matrices. This talk is meant to give an introduction to the theory of Jacobi matrices and orthogonal polynomials.

Rota's Universal Operators and Invariant Subspaces in Hilbert Spaces

Rota showed, in 1960, that there are operators $T$ that provide models for every bounded linear operator on a separable, infinite dimensional Hilbert space, in the sense that given an operator $A$ on such a Hilbert space, there is $\lambda \ne 0$ and an invariant subspace $M$ for $T$ such that the restriction of $T$ to $M$ is similar to $\lambda A$. In 1969, Caradus provided a practical condition for identifying such universal operators. In this talk, we will use the Caradus theorem to exhibit a new example of a universal operator and show how it can be used to provide information about invariant subspaces for Hilbert space operators. Of course, Toeplitz operators and composition operators on the Hardy space ${H}^{2}\left(𝔻\right)$ will play a role!

This talk describes work in collaboration with Eva Gallardo-Gutiérrez, Universidad Complutense de Madrid, done there this year during the speaker's sabbatical.

The Brownian traveller on manifolds

We study the inﬂuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the ﬂat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the ﬂat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

1. Martin Kolb and David Krejcirik: The Brownian traveller on manifolds, J. Spectr. Theory, to appear; preprint on arXiv:1108.3191 [math.AP].

Berezin Calculus over Weighted Bergman Spaces of Polyanalytic type

Starting from the Poincaré metric $d{s}^{2}=\frac{1}{2\pi i}{\left(1-\mid z{\mid }^{2}\right)}^{-2}d\stackrel{‾}{z}\phantom{\rule{thickmathspace}{0ex}}dz$ on the the unit disk $𝔻$, we will study the range of the Berezin transforms generated from the normalized kernel function ${K}_{\zeta }^{n}\left(z\right)={K}^{n}\left(z,\zeta \right){K}^{n}\left(\zeta ,\zeta {\right)}^{-\frac{1}{2}}$ regarding the weighted polyanalytic Bergman spaces ${A}_{n}^{\alpha }\left(𝔻\right)$ of order $n$. Special emphasize will be given to the invariance of the range of the Berezin transformation under the action of the Möbius transformations ${\phi }_{\zeta }\left(z\right)=\frac{z-a}{1-\stackrel{‾}{\zeta }z}$. Connection between Berezin calculus over weighted Bergman spaces of polyanalytic type on the disk $𝔻$ and on the upper half space ${ℂ}^{+}$ will also be discussed along the talk.

Noncommutative summands of the ${C}^{*}$-algebra ${C}_{r}^{*}{\mathrm{SL}}_{2}\left({𝔽}_{2}\left(\left(\varpi \right)\right)\right)$

Let ${𝔽}_{2}\left(\left(\varpi \right)\right)$ denote the Laurent series in the indeterminate $\varpi$ with coefficients over the finite field with two elements ${𝔽}_{2}$. This is a local nonarchimedean field with characteristic $2$. We show that the structure of the reduced group ${C}^{*}$-algebra of ${\mathrm{SL}}_{2}\left({𝔽}_{2}\left(\left(\varpi \right)\right)\right)$ is determined by the arithmetic of the ground field. Specifically, the algebra ${C}_{r}^{*}{\mathrm{SL}}_{2}\left({𝔽}_{2}\left(\left(\varpi \right)\right)\right)$ has countably many noncommutative summands, induced by the Artin-Schreier symbol. Each noncommutative summand has a rather simple description: it is the crossed product of a commutative ${C}^{*}$-algebra by a finite group. The talk will be elementary, starting from the scratch with the definition of ${C}_{r}^{*}{\mathrm{SL}}_{2}$.

A light introduction to supersymmetry

We give a brief introduction to supersymmetric quantum mechanics.

A Riemann-Hilbert approach to Toeplitz operators and the corona theorem

Together with differential operators, Toeplitz operators (TO) constitute one of the most important classes of non-self adjoint operators , and they appear in connection with various problems in physics and engineering. The main topic of my presentation will be the interplay between TOs and Riemann-Hilbert problems (RHP), and the relations of both with the corona theorem. It has been shown that the existence of a solution to a RHP with $2×2$ coefficient $G$, satisfying some corona type condition, implies – and in some cases is equivalent to – Fredholmness of the TO with symbol $G$. Moreover, explicit formulas for an appropriate factorization of $G$ were obtained, allowing to solve different RHPs with coefficient $G$, and to determine the inverse, or a generalized inverse, of the TO with symbol $G$. However, those formulas depend on the solutions to 2 meromorphic corona problems. These solutions being unknown or rather complicated in general, the question whether the factorization of $G$ can be obtained without the corona solutions is a pertinent one. In some cases, it already has a positive answer; how to solve this question in general is open, and all the more so in the case of $n×n$ matrix functions $G$, for which the results regarding the $2×2$ case have recently been generalized.

Generalized invertibility in rings: some recent results

The theory of generalized inverses has its roots both on semigroup theory and on matrix and operator theory. In this seminar we will focus on the study of the generalized inverse of von Neumann, group, Drazin and Moore-Penrose in a purely algebraic setting. We will present some recent results dealing with the generalized inverse of certain types of matrices over rings, emphasizing the proof techniques used.

Spectral analysis of some non-self-adjoint operators

We give an introduction to the study of one particular class of non-self-adjoint operators, namely $𝒫𝒯$-symmetric ones. We explain briefly the physical motivation and describe the classes of operators that are considered. We explain relations between the operator classes, namely their non-equivalence, and mention open problems.

In the second part, we focus on the similarity to self-adjoint operators. On the positive side, we present results on one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. Using functional calculus, closed formulas for the similarity transformation and the similar self-adjoint operator are derived in particular cases. On the other hand, we analyse the imaginary cubic oscillator, which, although being $𝒫𝒯$-symmetric and possessing real spectrum, is not similar to any self-adjoint operator. The argument is based on known semiclassical results.

1. P. Siegl: The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry, PRAMANA-Journal of Physics, Vol. 73, No. 2, 279-287,
2. D. Krejcirík, P. Siegl and J. Zelezný: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Analysis and Operator Theory, to appear,
3. P. Siegl and D. Krejcirík: On the metric operator for imaginary cubic oscillator, Physical Review D, to appear.

Corona conditions and symbols with a gap around zero

Convolution equations on a finite interval (which we can assume to be $\left[0,1\right]$) lead to the problem of factorizing matrix functions $G=\left[\begin{array}{cc}{e}_{-1}& 0\\ g& {e}_{1}\end{array}\right]$ where ${e}_{\theta }\left(\xi \right)={e}^{i\theta \xi }$, $\theta \in ℝ$ and $g\in {L}_{\infty }\left(ℝ\right)$. Here we consider $g$ of the form $g={a}_{+}{e}_{\mu }+{a}_{-}{e}_{-\sigma }$ with ${a}_{±}\in {H}_{\infty }\left({ℂ}^{±}\right)$ and $\mu ,\sigma >0$. Imposing some corona-type conditions on ${a}_{±}$, we show that solutions to the Riemann-Hilbert problem $G{h}_{+}={h}_{-}$, with ${h}_{±}\in \left({H}_{\infty }\left({ℂ}^{±}\right){\right)}^{2}$, can be determined explicitly and conditions for invertibility of the Toeplitz operator with symbol $G$ in $\left({H}_{p}^{+}{\right)}^{2}$ can be derived from them.

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Seminar organized in the context of the project PTDC/MAT/121837/2010.