Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2024 | Book

Weighted Hardy Spaces Over the Unit Ball: The Freely Noncommutative and Commutative Settings Read chapter Read first chapter

Operator and Matrix Theory, Function Spaces, and Applications

International Workshop on Operator Theory and its Applications 2022, Kraków, Poland

Editors: Marek Ptak, Hugo J. Woerdeman, Michał Wojtylak

Publisher: Springer Nature Switzerland

Book Series : Operator Theory: Advances and Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

This volume features presentations from the International Workshop on Operator Theory and its Applications that was held in Kraków, Poland, September 6-10, 2022. The volume reflects the wide interests of the participants and contains original research papers in the active areas of Operator Theory. These interests include weighted Hardy spaces, geometry of Banach spaces, dilations of the tetrablock contractions, Toeplitz and Hankel operators, symplectic Dirac operator, pseudodifferential and differential operators, singular integral operators, non-commutative probability, quasi multipliers, Hilbert transform, small rank perturbations, spectral constants, Banach-Lie groupoids, reproducing kernels, and the Kippenhahn curve. The volume includes contributions by a number of the world's leading experts and can therefore be used as an introduction to the currently active research areas in operator theory.

Table of Contents

Frontmatter
Weighted Hardy Spaces Over the Unit Ball: The Freely Noncommutative and Commutative Settings

It is known that backward-shift-invariant subspaces of the Hardy space \(H^2\) serve as the model spaces for a large class of contraction operators while forward-shift-invariant spaces (i.e., the orthogonal complements of backward-shift-invariant subspaces) have a related Beurling representation in terms of inner functions. Furthermore any such orthogonal decomposition of the whole space \(H^2\) also has a discrete-time linear-system interpretation. Recently there has been a surge of research activity elaborating on these and related results in multivariable settings, both commutative and freely noncommutative. Here we review these results with special emphasis on how additional perspective is had by looking at the commutative and freely noncommutative cases together.

Joseph A. Ball, Vladimir Bolotnikov
Two Aspects of Small Diameter Properties
Abstract
In this short note, we study two different geometrical aspects of Banach spaces with small diameter properties, namely the Ball Dentable Property (BDP), Ball Huskable Property (BHP) and Ball Small Combination of slice Property (BSCSP). We show that BDP, BHP and BSCSP are separably determined properties. We also explore the stability of these properties over Köthe-Bochner spaces.
Sudeshna Basu, Susmita Seal
Hilbert Transform in the Cartwright–de Branges Space
Abstract
Contrary to celebrated transforms such as the Fourier transform, explicit formulas for the Hilbert transform of well-known functions are rare. In this note, we present a formula for the Hilbert transform of \(\log |F/E|\), where F belongs to the Cartwright–de Branges space associated with the de Branges function E. The formula implies several other known results.
Arun K. Bhardwaj, Arup Chattopadhyay, Javad Mashreghi, R. K. Srivastava
A Note on the Dilation of a Certain Family of Tetrablock Contractions
Abstract
We find an explicit tetrablock isometric dilation for every member \((A_\alpha , B, P)\) of a family of tetrablock contractions indexed by a parameter \(\alpha \) in the closed unit disc (only the first operator of the tetrablock contraction depends on the parameter). The dilation space is the same for any member of the family. Explicit dilation for the adjoint tetrablock contraction \((A_\alpha ^*, B^*, P^*)\) for every member of the family mentioned above is constructed as well. This example is important because it has been claimed in the literature that this example does not have a dilation. Taking cue from this construction and using Toeplitz operators on \(H^2_{\mathbb D}(\mathcal D_P)\), we obtain necessary and sufficient conditions for a tetrablock contraction to have a certain type of tetrablock isometric dilation.
Tirthankar Bhattacharyya, Mainak Bhowmik
Commuting Toeplitz Operators and Moment Maps on Cartan Domains of Type III
Abstract
Let \(D^{III}_n\) and \(\mathcal {S}_n\) be the Cartan domains of type III that consist of the symmetric \(n \times n\) complex matrices Z that satisfy \(Z\overline {Z} < I_n\) and \(\mathrm {Im}(Z) > 0\), respectively. For these domains, we study weighted Bergman spaces and Toeplitz operators acting on them. We consider the Abelian groups \(\mathbb {T}\), \(\mathbb {R}_+\) and \(\mathrm {Symm}(n,\mathbb {R})\) (symmetric \(n \times n\) real matrices), and their actions on the Cartan domains of type III. We call the corresponding actions Abelian Elliptic, Abelian Hyperbolic and Parabolic. The moment maps of these three actions are computed and functions of them (moment map symbols) are used to construct commutative \(C^*\)-algebras generated by Toeplitz operators. This leads to a natural generalization of known results for the unit disk. We also compute spectral integral formulas for the Toeplitz operators corresponding to the Abelian Elliptic and Parabolic cases.
David Cuevas-Estrada, Raul Quiroga-Barranco
Branching Symplectic Monogenics Using a Mickelsson–Zhelobenko Algebra
Abstract
In this paper we consider (polynomial) solution spaces for the symplectic Dirac operator (with a focus on 1-homogeneous solutions). This space forms an infinite-dimensional representation space for the symplectic Lie algebra \(\mathfrak {sp}(2m)\). Because \(\mathfrak {so}(m)\subset \mathfrak {sp}(2m)\), this leads to a branching problem which generalises the classical Fischer decomposition in harmonic analysis. Due to the infinite nature of the solution spaces for the symplectic Dirac operators, this is a non-trivial question: both the summands appearing in the decomposition and their explicit embedding factors will be determined in terms of a suitable Mickelsson–Zhelobenko algebra.
David Eelbode, Guner Muarem
On Non-commutative Spreadability
Abstract
We review some results on spreadable quantum stochastic processes and present the structure of some monoids acting on the index-set of all integers \({\mathbb Z}\). These semigroups are strictly related to spreadability, as the latter can be directly stated in terms of invariance with respect to their action.
We are mainly focused on spreadable, Boolean, monotone, and q-deformed processes. In particular, we give a suitable version of the Ryll-Nardzewski Theorem in the aforementioned cases.
Maria Elena Griseta
Small Rank Perturbations of H-Expansive Matrices
Abstract
In this paper small rank perturbations of H-expansive and H-unitary matrices are explored. Particular attention is given to the location of eigenvalues with respect to the unit circle for these classes of matrices. The canonical form (in the H-unitary case) and the simple form (in the H-expansive case) for the pair \((A,H)\) will be the starting point.
G. J. Groenewald, D. B. Janse van Rensburg, A. C. M. Ran
On the Berger-Coburn Phenomenon
Abstract
In their previous work, the authors proved the Berger-Coburn phenomenon for compact and Schatten \(S_p\) class Hankel operators \(H_f\) on generalized Fock spaces when \(1<p<\infty \), that is, for a bounded symbol f, if \(H_f\) is a compact or Schatten class operator, then so is \(H_{\bar f}\). More recently J. Xia has provided a simple example that shows that there is no Berger-Coburn phenomenon for trace class Hankel operators on the classical Fock space \(F^2\). Using Xia’s example, we show that there is no Berger-Coburn phenomena for Schatten \(S_p\) class Hankel operators on generalized Fock spaces \(F^2_\varphi \) for any \(0<p\le 1\). Our approach is based on the characterization of Schatten class Hankel operators while Xia’s approach is elementary and heavily uses the explicit basis vectors of \(F^2\), which cannot be found for the weighted Fock spaces that we consider. We also formulate four open problems.
Zhangjian Hu, Jani A. Virtanen
Quasi-Multipliers and Algebrizations of an Operator Space. III
Abstract
The main part of this article serves as a corrigendum to the author’s paper “Quasi-multipliers and algebrizations of an operator space. J. Funct. Anal. 251(1):346–359 (2007).” We also present recreational examples in the form of a quiz which illustrates the main theorem of the aforementioned paper. Furthermore, we give the answer to one of the open questions raised by the author.
Masayoshi Kaneda
Mellin Pseudodifferential Operators and Singular Integral Operators with Complex Conjugation
Abstract
The paper is devoted to studying Banach algebras \({\mathfrak B}_{SO}\) and \({\mathfrak B}_{QC}\) of singular integral operators with complex conjugation and slowly oscillating (SO) or quasicontinuous (QC) coefficients on weighted Lebesgue spaces over star-like curves \(\Gamma \) without cusps by applying algebras of Mellin pseudodifferential operators with non-regular symbols. Making use of Mellin pseudodifferential operators with slowly oscillating \(V(\mathbb {R})\)-valued symbols of limited smoothness on \(\mathbb {R}_+\) or with quasicontinuous \(V(\mathbb {R})\)-valued symbols on \(\mathbb {R}_+\), where \(V(\mathbb {R})\) is the Banach algebra of absolutely continuous functions of bounded total variation on \(\mathbb {R}\), we construct a Fredholm symbol map for the Banach algebra \({\mathfrak B}_{SO}\) and obtain a Fredholm criterion and an index formula for the operators \(T\in {\mathfrak B}_{SO}\), and construct a Fredholm symbol map for the Banach algebra \({\mathfrak B}_{QC}\) and establish a Fredholm criterion for the operators \(T\in {\mathfrak B}_{QC}\).
Yuri I. Karlovich, Francisco J. Monsiváis-González
Maximal Noncompactness of Singular Integral Operators on Spaces with Some Khvedelidze Weights
Abstract
Let \(\Gamma \) be a contour in the complex plane consisting of a finite number of circular arcs joining the endpoints \(-1\) and 1, possibly including the segment \([-1,1]\). We consider the singular integral operator \(A=aI+bS_\Gamma \) with constant coefficients \(a,b\in \mathbb {C}\), where \(S_\Gamma \) is the Cauchy singular integral operator over \(\Gamma \). We provide a detailed proof of the maximal noncompactness of the operator A on \(L^2\) spaces with the Khvedelidze weights \(\varrho (t)=|t-1|{ }^\beta |t+1|{ }^{-\beta }\) satisfying \(-1<\beta <1\). This result was announced by Naum Krupnik in 2010, but its proof has never been published.
Oleksiy Karlovych, Alina Shalukhina
On the Dual Representation of the Congruence Kernels and the Related Delsarte Type Transmutations of Multidimensional Differential Operators
Abstract
We analyze a dual representation of the congruence operator kernels subject to a pair of multidimensional differential operators on a Hilbert space within both the spectral operator theory and Volterrian projector operator calculus. The related Volterrian type factorization of Fredholm operators is effectively solved for the case of special trace class elliptic operators. Based on the existence of operator kernels, spectrally congruent to the related pairs of operators, they are constructed by means of two approaches, using both of the kernel representation of the spectral operators in related Hilbert spaces and of the factorization property of Fredholm operators within the Volterrian projector operator calculus. In case of the multidimensional trace-class operator valued algebra of pseudo-differential expressions we stated that the corresponding kernel of a Fredholm operator, factorized by means of Volterrian kernel operators and commuting to a given elliptic representative, coincides with its fractional power. Application to construction of the related Delsarte type transmutation operators is presented.
Anatolij K. Prykarpatski, Petro Y. Pukach, Myroslava I. Vovk
Toeplitz Operator with a Finite Number of Horizontal Symbols Acting on the Poly-Fock Spaces
Abstract
We study the \(C^*\)-algebra generated by Toeplitz operators with extended horizontal symbols acting on the Poly-Fock spaces of the complex plane. We show that this \(C^*\)-algebra coincides with the algebra generated by a finite number of Toeplitz operators with horizontal symbols, without the need for the explicit calculation of the spectral functions.
Armando Sánchez-Nungaray, Carlos González-Flores, Miguel Antonio Morales-Ramos, María del Rosario Ramírez-Mora
On Abstract Spectral Constants
Abstract
We prove bounds for a class of homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix–Palencia, Okubo–Ando and von Neumann in a unified way and to refine a recent result by Crouzeix–Greenbaum.
Felix L. Schwenninger, Jens de Vries
Geometric Structures Related to a -Algebra
Abstract
In this presentation we will show that the groupoid of partially invertible elements and, in particular, the groupoid of partial isometries of a \(W^*\)-algebra, have the structure of Banach- Lie groupoids. The relationship between these groupoids, their algebroids and the Banach-Poisson geometry will also be shown.
Aneta Sliżewska
Composition in Reproducing Kernel Hilbert Spaces à Rebours
Abstract
This is a recapitulation of the talk (under the same title) designated for IWOTA2022 Session 21 “Truncated and Full Moment Problems, and Applications”.
Roughly, the content of this paper provides us with building up the univocal treatment of extending positive definiteness forward and backward. The paper is completed with exposing vital facts (in two Appendices) concerning reproducing kernel Hilbert spaces, the main ingredient of it.
Franciszek Hugon Szafraniec
Kippenhahn’s Construction Revisited
Abstract
Kippenhahn discovered that the numerical range of a complex square matrix is the convex hull of a plane real algebraic curve. Here, we present an example of a convex set, which has a similar algebraic description as the numerical range, whereas the analogue of Kippenhahn’s construction fails regarding isolated, singular points of the curve. This example prompted us to carefully review Kippenhahn’s assertion and to highlight aspects of a complete proof that was achieved with methods of convex geometry and real algebraic geometry.
Stephan Weis
On de Finetti-Type Theorems
Abstract
The investigation of distributional symmetries was initiated by de Finetti’s celebrated theorem, which shows that any finite joint-distribution of sequences of two-point valued exchangeable random variables is obtained by randomization of the binomial distribution. This result has since found several generalizations both in classical and noncommutative settings. In this paper, we discuss a series of recent results that extend de Finetti’s theorem to various non-commutative models, including Bolean, monotone, CAR algebra as well as \(\mathbb {Z}_2\)-graded \(C^*\)-algebras.
Paola Zurlo
Metadata
Title
Operator and Matrix Theory, Function Spaces, and Applications
Editors
Marek Ptak
Hugo J. Woerdeman
Michał Wojtylak
Copyright Year
2024
Electronic ISBN
978-3-031-50613-0
Print ISBN
978-3-031-50612-3
DOI
https://doi.org/10.1007/978-3-031-50613-0

Premium Partner

    Image Credits