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

- Benali Abdelkader

University of Chlef, Algeria ## Generalizations of Kaplansky’s Theorem Involving Unbounded Linear Operators

The purpose of this paper is to generalize a very famous result on products of normal operators, due to I. Kaplansky. The context of generalization is that of bounded hyponormal and unbounded normal operators on1 complex separable Hilbert spaces. Some examples "spice up" the paper. enough so that even monographs have been devoted to them. In this paper we are mainly interested in generalizing the following result to unbounded normal and bounded hyponormal operators.

- Maria Stella Adamo

Università degli Studi di Roma "Tor Vergata" ## Representable and continuous functionals on Banach quasi *-algebras and applications to tensor products

Representable functionals provide an essential tool to study the structural properties of a Banach quasi *-algebra. In particular, these functionals play a central role in investigating the existence of *-representations of the quasi *-algebra.

A particular class of representable functionals is constituted by those that are continuous. It is not known in general if representable functionals are automatically continuous. Nonetheless, an affirmative answer has been given under specific hypotheses, verified in the case of $(L^2(I,d\lambda),\mathcal{C}(I))$ and $(L^2(I,d\lambda),L^{\infty}(I))$, for $I=[0,1]$ endowed with the Lebesgue measure $\lambda$.

In this talk, I will present results about the continuity of representable functionals and the applications they have on the study of properties for tensor products of Banach quasi *-algebras.

The first part of this talk is joint work with C. Trapani (University of Palermo - Italy). The second part is joint work with M. Fragoulopoulou (National and Kapodistrian University of Athens - Greece).

- Il Ju An

Ewha Womans University ## Krein space adjoint operators and their Aluthge transform

We first study the polar decomposition of a Krein space selfadjoint operator and the relation between the $(s,t)$-Aluthge transform and the class of Krein space selfadjoint operators. We explore various spectra of a Krein space adjoint operator and its Aluthge transform. Also we introduce some examples of Krein space selfadjoint operators satisfying the conditions. Furthermore, we discuss when the product of an operator and its Krein space adjoint operator becomes $\mathcal J$-Fredholm, $\mathcal J$-Weyl, or $\mathcal J$-Browder.

- Fernanda Botelho

University of Memphis, USA ## Projections on spaces of vector valued continuous functions

In this talk I explain some aspects of the structure of the ranges of bi-contractive projections on spaces of vector valued continuous functions. A crucial tool is a vector valued Tietze Extension Theorem, derived from results on M-ideals.

This talk includes results from joint work with T. S. S. R. K. Rao.

- Zbigniew Burdak

University of Agriculture ## On joint invariant subspaces under pairs of commuting isometries.

Joint invariant subspaces under a pair of (commuting) isometries may be investigated via joint invariant subspaces under an extension to another pair of (commuting) isometries. We show that a general pair of isometries, for any relatively prime, positive integers $m,n$ has a minimal extension to a pair of the form $(U^kV^n ,U^lV^m)$ where $U$ is a unitary operator commuting with an isometry $V$ and $km − ln = 1,0 \lt k \lt n$, $0 \leq l \lt m$. We present the model of pairs of the form $(U^kV^n ,U^lV^m)$ and investigate their joint invariant subspaces by invariant subspaces of $V$. Especially interesting cases are if $V$ is a unilateral shift.

- Raúl E. Curto

University of Iowa ## The Extended Aluthge Transform

Given a bounded linear operator $T$ with canonical polar decomposition $T=V|T|$, the Aluthge transform of $T$ is the operator $\Delta(T):=|T|^{1/2} V |T|^{1/2} $. For $P$ an arbitrary positive operator such that $VP=T$, we define the extended Aluthge transform of $T$ associated with $P$, denoted by $\Delta_p(T):= P^{1/2} V P^{1/2} $.

First, we establish some basic properties of $\Delta_p $; second, we study the fixed points of the extended Aluthge transform; third, we consider the case when $T$ is an idempotent; next, we discuss whether $\Delta_p $ leaves invariant the class of complex symmetric operators.

We also study how $\Delta_p $ transforms the numerical radius and numerical range. As a key application, we prove that the spherical Aluthge transform of a commuting pair of operators corresponds to the extended Aluthge transform of a $2\times 2 $ operator matrix built from the pair; thus, the theory of extended Aluthge transforms yields results for spherical Aluthge transforms.

The talk is based on joint work with Chafiq Benhida.

- Adam Dor-On

University of Illinois at Urbana-Champaign, USA ## Self-adjoint irreversibly generated graph algebras

The Toeplitz-Cuntz-Krieger algebra of a directed graph is the $C^\ast$-algebra generated by concatenation operators on square summable sequences indexed by finite paths of the graph. Its smallest gauge-invariant quotient is the celebrated Cuntz-Krieger algebra, which is deeply connected to the associated subshift of finite type and automata of the directed graph.

Understanding representations of such Toeplitz-Cuntz-Krieger algebras turns out to be useful for producing wavelet on Cantor sets by Marcolli and Paolucci and in the study of semi-branching function systems by Bezuglyi and Jorgensen. Together with Davidson and Li, we provided a non-self-adjoint perspective for such representations, which led to new invariants that distinguish them up to unitary equivalence.

In this talk I will present a complete characterization of those finite directed graphs that admit weakly-closed self-adjoint algebras that are generated only by represented concatenation operators (without their adjoints !). The first example of this counter-intuitive phenomenon was produced by Read in the case where the graph has a single vertex and two loops. I will explain how the periodic Road Coloring theorem of Béal and Perrin from automata theory is used to promote Read’s example to all finite non-cycle graphs.

Based on joint work with Christopher Linden.

- Daniel Girela

Universidad de Málaga, Spain ## Multipliers and integration operators between conformally invariant spaces

In this talk we shall be concerned with two classes of conformally invariant spaces of analytic functions in the unit disc $\mathbb D$, the Besov spaces $B^p$ $(1\le p\lt \infty )$ and the $Q_s$ spaces $(0\lt s \lt \infty)$.

The talk is based on a joint work with Noel Merchán.

- Andreas Hartmann

Université de Bordeaux ## Random interpolating sequences in Dirichlet spaces

In this talk I will discuss random interpolating sequences in weighted Dirichlet spaces $\mathcal{D}_\alpha$, $0\leq \alpha\leq 1$. The results in particular imply that almost sure interpolating sequences for $\mathcal{D}_\alpha$ are exactly the almost sure separated sequences when $0\le \alpha \lt 1/2$ (which includes the Hardy space $H^2=\mathcal{D}_0$), and they are exactly the almost sure zero sequences for $\mathcal{D}_\alpha$ when $1/2\lt \alpha \lt 1$. I will also discuss the situation in the classical Dirichlet space $\mathcal{D}=\mathcal{D}_1$ where we get an almost optimal result.

The results are inspired by work by Cochran and by Rudowicz who considered random interpolating sequences in Hardy spaces based on separation and Carleson measure conditions. In Hardy spaces, the condition characterizing almost sure separation implies almost surely the Carleson measure condition which gives the result on almost surely interpolating sequences. As a motivation for our results in Dirichlet spaces, I show that already in Hardy spaces there are sequences which satisfy almost surely the Carleson measure condition without giving almost sure separation improving thereby Rudowicz’ result.

This is joint work with Nikolaos Chalmoukis, Karim Kellay and Brett Wick.

- Alexander Helemskii

Moscow State (Lomonosov) University, Russia ## A general frame-work for the $p$-convex tensor product of multinormed spaces

A. Lambert introduced a new type of structures, called by him sequence spaces, that were, in a sense, intermediate between classical normed spaces and operator spaces. One of the main achievements of his theory was the existence theorem for tensor products of his spaces. Lambert spaces had as a base space, $\ell_2$. After the appearance, in papers of Dales and other mathematicians, of the so-called $p$-multi-normed spaces, based on $\ell_p; 1\le p\le\infty$; the speaker had the suspicion that the theorem of Lambert can be extended to a much wider class of base spaces, including $L_p(X,\mu)$, with a continuous measure, $\ell_p$ and a lot of others.

Indeed, we suggest such a class: these are the so-called convenient stratified spaces, The main theorem of this talk establishes the existence of the tensor product of $p$-convex spaces having as a base the previously mentioned space; this extends the theorem of Lambert in several directions. We shall formulate this theorem and discuss highlights of the proof.

- Dijana Ilisevic

University of Zagreb, Croatia ## An inverse eigenvalue problem for isometries

This talk is related to the following inverse eigenvalue problem for isometries: when is a given finite set of modulus one complex numbers the spectrum of a surjective linear isometry? Necessary conditions on such a set will be presented. The problem of determining sufficient conditions seems to be much more complicated and related to the structure of specific Banach spaces. A particular emphasis will be given to $C_0(\Omega)$, the Banach space of continuous complex-valued functions on a connected locally compact Hausdorff space $\Omega$ vanishing at infinity.

The recent results that will be presented in this talk are from joint work with Fernanda Botelho (University of Memphis, USA) and joint work with Chih-Neng Liu and Ngai-Ching Wong (National Sun Yat-sen University, Taiwan). The work of Dijana Ilisevic has been fully supported by the Croatian Science Foundation under the project IP-2016-06-1046.

- Joanna Jurasik

Maria Curie-Sklodowska University, Lublin, Poland ## Asymmetric truncated Toeplitz operators on infinite-dimensional spaces

Let $H^2$ denote the standard Hardy space in the open unit disk $\mathbb{D}$ and let $\mathbb{T}=\partial\mathbb{D}$. With any nonconstant inner function $\alpha$ we associate the model space $K_{\alpha}=H^2\ominus \alpha H^2$. Truncated Toeplitz operators are compressions of classical Toeplitz operators to model spaces. We consider their generalizations, the so-called asymmetric truncated Toeplitz operators. Let $\alpha,\beta$ be two inner functions and $\varphi\in L^2(\mathbb{T})$. An asymmetric truncated Toeplitz operator $A^{\alpha,\beta}_{\varphi}$ is the operator from $K_{\alpha}$ into $K_{\beta}$ given by $$A^{\alpha,\beta}_{\varphi}=P_{\beta}(\varphi f), \qquad f\in K_{\alpha}\cap H^{\infty},$$ where $P_{\beta}$ is the orthogonal projection from $L^2(\mathbb{T})$ onto $K_{\beta}$.

In this talk we present some properties of asymmetric truncated Toeplitz operators on infinite-dimensional model spaces. In particular, we present their characterizations in terms of matrix representations with respect to some natural bases.

- Dong-O Kang

Chungnam National University ## Some properties of Block Hankel operator kernels

Some of the recent and interesting results on kernels of Block Hankel operators will be presented. The focus will be on the relation between inner functions associated with the Hankel operator kernels and the symbol functions.

- Alexander Kheifets

University of Massachusetts Lowell ## Hardy Spaces of Fuchsian Groups in the Upper Half-plane

We establish exact conditions for non triviality of all subspaces of the Hardy space (with respect to the Lebesgue measure) in the upper half plane, that consist of the character automorphic functions with respect to the action of a discrete subgroup of $SL_2(\mathbb R)$. Such spaces are the natural objects in the context of the spectral theory of almost periodic differential operators . It is parallel to the celebrated Widom-Pommerenke characterization for Hardy spaces (with respect to the harmonic measure) with the following modification: the Green function of the group is substituted with the Martin function and also the Martin measure must be a pure point one.

- Eungil Ko

Ewha Womans University, Korea ## Remark on truncated Toeplitz operators

In this paper, we study properties of an operator $S_{\phi,\psi}^u\in L(L^2)$ defined by \[S_{\phi,\psi}^uf=\phi P_uf+\psi Q_uf, f\in L^2,\] for two symbols $\phi, \psi\in L^{\infty}$ and an inner function $u$, where $Pu$ denotes the orthogonal projection of $L^2$ onto the model space $K^2_u:=H^2 \theta uH^2$ and $Q_u$ denotes the orthogonal projection of $L^2$ onto $(K^2_u)^\perp$.

In particular, we provide necessary and sufficient conditions for the operator $S_{\phi,\psi}^u$ to be nonnegative and self-adjoint. Finally, we consider the symbols $\phi$ and $\psi$ when $S_{\phi,\psi}^u$ is normal.

### References

- J. Agler and J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, 44, Amer. Math. Soc., 2002.
- A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine angew. Math., 213(1963-1964), 89-102.
- A. Böttcher and B. Silbermann, Analysis of Toeplitz operators, Springer Monographs in Mathematics, 2012.
- I. Chalendar and D. Timotin, Commutation relation for Truncated Toeplitz operators, Oper. Matrices, 8(2014), 877-888.
- Joseph A. Cima, William T. Ross, and Warren R. Wogen, Truncated Toeplitz operators on ﬁnite dimensional spaces, Oper. Matrices, 3(2008), 357-369.
- P. L. Duren, Theory of $H^p$ spaces, Dover, Pub., 2000.
- S. R. Garcia, Aluthge transforms of complex symmetric operators, Int. Eq. Op. Th. 60(2008), 357-367.
- S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc., 358(2006), 1285-1315.
- S. R. Garcia and W. T. Ross, Recent progress on truncated Toeplitz operators, Blaschke Products and Their Applications, Fields Inst. Commun., 65(2013), 275-319.
- P. R. Halmos, A Hilbert space problem book, Second edition, Springer-Verlag, New York, 1982.
- P. R. Halmos,Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970), 887-933.
- E. Ko and J. Lee, Normal truncated Toeplitz operators on ﬁnite dimensional spaces, Linear and Multilinear Algebra, 63(10)(2015), 1947-1971.
- E. Ko, J. Lee, and T. Nakazi, On the Dilation of Truncated Toeplitz Operators II, Complex Anal. Op. Th., to appear.
- E. Ko, J. Lee, and T. Nakazi, Hyponormality of the Dilation of Truncated Toeplitz Operators, preprint.
- S. Nagy, C. Foias, H. Bercovici, and L. Kerchy, Harmonic analysis of operators on Hilbert space, Springer, 2010.
- T. Nakazi, Invariant subspaces of Toeplitz operators and uniform algebras, Bull. Belg. Math. Soc. Simon Stevin 15(2008), 1-8.
- T. Nakazi and T. Yamamoto, Normal singular integral operators with Cauchy kernel on L2, Int. Eq. Op. Th., 78(2)(2014), 233-248.
- D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices, 1(2007), 419-526.
- N. A. Sedlock, Algebras of truncated Toeplitz operators, Oper. Matrices, 5(2011), 309-326.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2016R1D1A1B03931937).

- David Krejcirik

Czech Technical University in Prague ## From functional inequalities to spectral properties of Schrödinger and Dirac operators

We shall present two independent ways of passing from functional inequalities of Hardy type to spectral properties of Schrödinger and Dirac operators with complex potentials. First, by applying the Birman-Schwinger principle, which reduces the spectral problem for the partial differential operator to an integral equation, we obtain sharp estimates of eigenvalues of Schrödinger and Dirac operators in one and three dimensions. Second, by developing the method of multipliers, we shall establish the total absence of eigenvalues for electromagnetic Schrödinger operators in all dimensions under various alternative hypotheses, still allowing complex-valued potentials with critical singularities. This is joint work with Luca Fanelli and Luis Vega.

- Bartosz Łanucha

Maria Curie-Skłodowska University, Lublin, Poland ## Conjugations on $L^2(\mathbb{T})$ preserving shift invariant subspaces of $H^2$

A conjugation on a Hilbert space is an antilinear isometric involution. In this talk we are interested in conjugations on $L^2(\mathbb{T})$. We consider conjugations commuting with the multiplication operator $M_z$ or intertwining $M_z$ and its adjoint $M_{\bar z}$. We describe which of these conjugations preserve subspaces invariant for the shift operator on $H^2$.

The talk is based on joint work with M. C. Câmara, K. Kliś-Garlicka and M. Ptak.

- Zinaida Lykova

Newcastle University ## Algebraic aspects of rational inner functions from the unit disc to the tetrablock

The set \[\overline{\mathbb{E}}= \{ x \in {\mathbb{C}}^3: 1-x_1 z - x_2 w + x_3 zw \neq 0 \text{ whenever }|z| \lt 1, |w| \lt 1 \}\] is called the tetrablock and has intriguing complex-geometric properties; it is a polynomially convex, nonconvex, starlike about $0$; it has a group of automorphisms parametrised by ${\mathrm{Aut~}} {\mathbb{D}} \times {\mathrm{Aut~}}{ \mathbb{D}} \times {\mathbb{Z}}_2$, its distinguished boundary is homeomorphic to the solid torus $\bar{\mathbb{D}} \times {\mathbb{T}}$. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc ${\mathbb{D}}$ to $\overline{\mathbb{E}}$ that map the boundary of the disc to the distinguished boundary $b\overline{\mathbb{E}}$ of $\overline{\mathbb{E}}$.

- Małgorzata Michalska

Maria Curie-Skłodowska University ## Characterizations of Asymmetric Truncated Toeplitz and Hankel Operators

Let $H^2$ be the Hardy space in the unit disk and for an inner function $\alpha$ let $K_\alpha=H^2\ominus\alpha H^2$. For $\psi\in L^2$, an asymmetric truncated Toeplitz operator $A^{\alpha,\beta}_\psi$ is defined on a dense subset $K_\alpha\cap H^{\infty}$ of $K_\alpha$ by $$A^{\alpha,\beta}_\psi=P_\beta(\psi f),$$ where $P_\beta$ is the orthogonal projection from $L^2$ onto $K_\beta$, and an asymmetric truncated Hankel operator $B^{\alpha,\beta}_\psi$ is defined on $K_\alpha\cap H^{\infty}$ by $$B^{\alpha,\beta}_\psi=P_\beta J(I-P)(\psi f),$$ where $P$ is the orthogonal projection from $L^2$ onto $H^2$ and $J$ is given by $J(z)=\overline{z}f(\overline{z})$ for $|z|=1$.

In this talk we present characterizations of asymmetric truncated Toeplitz operators in terms of compressed shifts and rank-two operators of special form. We also show a connection between asymmetric truncated Toeplitz operators and asymmetric truncated Hankel operators. We then use this connection to generalize results known for truncated Hankel operators to the asymmetric case.

This talk is based on joint work with C. Gu and B. Łanucha.

- Miguel Monsalve

Universidad Complutense de Madrid ## Spectral decomposition arising from Atzmon

In this talk, we will review Atzmon's Theorem on hyperinvariant subspaces for linear bounded operators acting on a separable Banach space which generalizes the well-known Wermer's Theorem. In particular, we will extend Atzmon's result to a more general framework and discuss some applications regarding Bishop operators.

This is a joint work with Eva A. Gallardo-Gutiérrez.

- Gabriel Kantun Montiel

Benemerita Universidad Autonoma de Puebla ## Fredholm theory relative to an homomorphism between Banach algebras

Fredholm theory arises in the context of the study of bounded linear operators in Banach or Hilbert spaces. By now, there are several results of Fredholm Theory, including generalizations for $C^\ast$-algebras, Banach algebras, or even rings. The underlying idea behind some of these generalizations is the invertibility of some elements of a ring with respect to some fixed ideal. Thus, some generalizations of Fredholm theory can be studied using generalized inverses. In this talk, we will deal with generalized inverses related to some spectral sets. We are particulary interested in generalizations of Fredholm theory with respect to a homomorphism between Banach algebras. Some applications include the study of Calkin algebras.

- Maria Nowak

Maria Curie-Sklodowska University ## On kernels of Toeplitz operators

Let $H^2$ denote the standard Hardy space on the unit disk $\mathbb D$ and let $\mathbb T=\partial \mathbb D$. For $\varphi\in L^{\infty}(\mathbb T)$ the Toeplitz operator on $H^2$ is given by $T_{\varphi}f=P_{+}(\varphi f)$, where $P_{+}$ is the orthogonal projection of $L^2(\mathbb T)$ onto $H^2$. It is known that the kernel of a $T_{\varphi} $ is a subspace of $H^2$ of the form $\text{Ker} T_{\varphi}= \gamma K_I$, where $K_I= H^2\ominus IH^2$ is the model space corresponding to the inner function $I$ such that $I(0)=0$ and $\gamma$ is an outer function of unit $H^2$ norm that acts as an isometric multiplier from $K_I$ onto $\gamma K_{I}$. Moreover, $\gamma$ can be expressed as $\gamma=\frac{a}{1-Ib_0}$, where $a$ and $b$ are functions from the unit ball of $H^{\infty}$ such that $|a|^2+|b_0|^2=1$ a.e. on $\mathbb T$ and $\left(\frac{a}{1-b_0}\right)^2 $ is a rigid function in $H^1$. In the recent paper [1], the authors considered the Toeplitz operator $T_{\frac{\overline{a}}{a}} $ where $a\in H^{\infty}$ is outer. Among other results, they described all outer functions $a$ such that $ \text{Ker} T_{\frac{\overline{a}}{a}}=K_{I}$. In the talk we describe all such functions $a$ for which $\text{ker}T_{\frac{\overline{a}}{a}}=\gamma K_{I}$. We also discuss properties of the kernels of some Toeplitz operators.

The talk is based on joint work with P. Sobolewski, A. Soltysiak and M. Woloszkiewicz-Cyll.

### Reference

- E. Fricain, A. Hartmann and W.T. Ross,
*Range spaces of co-analytic Toeplitz operators*, Canad. J. Math.**70**(2018) no. 6, 1261-1283.

- E. Fricain, A. Hartmann and W.T. Ross,
- Jesús Ocáriz

Universidad Autónoma de Madrid ## On the dual of variable Lebesgue spaces

Given an open interval $I \subseteq \mathbb{R}$ and a measurable function $p:I\to [1,+\infty)$, the variable Lebesgue space $L^{p(\cdot)}(I)$ is the subspace of measurable functions $f:I\to \mathbb{R}$ such that the following norm is finite \[\Vert f \Vert_{L^{p(\cdot)}(I)}=\inf\left\{\lambda>0: \int_I \left|\frac{f(x)}{\lambda}\right|^{p(x)} \mathrm{d}x\leq 1\right\}.\] The topological dual of this Banach space is perfectly known when $\Vert p\Vert_{L^\infty(I)}\lt \infty$. However, if $ \Vert p\Vert_{L^\infty(I)}=\infty $, describing it has been an open problem for years. In this talk, we are going to discuss some recent approaches that give a better understanding of the phenomena beyond it. Joint work with A. Amenta (University of Bonn), J. M. Conde-Alonso (Universidad Autónoma de Madrid) and D. Cruz-Uribe (University of Alabama).

- Lourdes Palacios

Universidad Autónoma Metropolitana- Iztapalapa ## On some algebras of functions with values in certain topological algebras

Let X be a completely regular Hausdorff space or a pseudocompact Hausdorff space. Let $C(X,A)$ be the algebra of all continuous functions on $X$ with values in a complex unital locally pseudo-convex algebra $A$. Let $C_b(X,A)$ be the subalgebra consisting of all bounded continuous functions, endowed with the topology given by the uniform pseudo-seminorms of $A$ on $X$. In this talk we will examine some properties of $A$ that are inherited by $C_b(X,A)$. These properties are related with projective limit decomposition, inversion, involution, spectral properties and metrizability. We will also outline an ongoing research considering more general algebras of functions.

- Alexei Poltoratski

Texas A&M University ## Toeplitz methods in spectral problems for canonical Hamiltonian systems

Recent approach based on use of truncated Toeplitz operators has produced new algorithms and examples in the area of inverse spectral problems for differential operators. In my talk I will discuss these results and discuss further questions. The talk is based on joint work with N. Makarov.

- Marek Ptak

University of Agriculture in Krakow, Krakow, Poland ## Conjugations in $L^2$ spaces of vector valued functions.

Antilinear, isometric, involutions — conjugations — in $L^2(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space, are considered. Those which commute with multiplication by independent variable $\mathbf{M}_z$ or intertwine $\mathbf{M}_z$ and $\mathbf{M}_{\bar z}$ are characterized. We also investigate which of them leave invariant the whole Hardy space $H^2(\mathcal{H})$. For a pure operator valued inner function $\Theta$ the subspace $\Theta H^2(\mathcal{H})$ is invariant for ${{\mathbf{M}_z}_{\mid H^2(\mathcal{H})}}$ and the model space $K_{\Theta}=H^2(\mathcal{H})\ominus\Theta H^2(\mathcal{H})$ is invariant for its adjoint $({{\mathbf{M}_z}_{\mid H^2(\mathcal{H})}})^*$. We investigate conjugations, which leave invariant also this spaces.

Joint work with M. C. Câmara, K. Kliś-Garlicka, B. Łanucha.

- William T. Ross

University of Richmond, USA ## Inner vectors on Hilbert spaces

Inspired by Beurling’s classical definition of an inner function relating to the shift on the Hardy space, we explore the other notions of inner vectors for general operators on Hilbert spaces.

- Ayse Sandikci

Ondokuz Mayis University, Faculty of Arts and Sciences, Department of Mathematics ## Multilinear $\tau $-Wigner Transform

In this work, we introduce a notation of multilinear $\tau $-Wigner transform.

We give a simple relation between short-time Fourier transform and multilinear $\tau $-Wigner transform. Further, we prove that multilinear $\tau $-Wigner transform is bounded on products of Lebesgue spaces. After we list some properties for multilinear $\tau $-Wigner transform. From these results we then prove the boundedness properties of multilinear $\tau $-Wigner transform on modulation spaces.

Some key references are given below.

- P. Boggiatto, G. De Donno, A. Oliaro: Time-frequency representations of Wigner type and pseudo-differential operators, Trans Amer Math Soc, 362 (2010), 4955-4981.
- A. Bényi, K. H. Gröchenig, C. Heil, K. A. Okoudjou: Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory, 54(2) (2005), 387-399.
- H. G. Feichtinger: Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, (1983).
- K. Gröchenig: Foundation of Time-Frequency Analysis, Birkhauser, 2001.
- A. Sandikci: Continuity of Wigner-type operators on Lorentz spaces and Lorentz mixed normed modulation spaces, Turk J Math., 38 (2014), 728-745.
- N. Teofanov: Continuity Properties of Multilinear Localization Operators on Modulation Spaces. In: Boggiatto P. et al. (eds) Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhauser, Cham, 2019.

- Andrzej Sołtysiak

Adam Mickiewicz University, Poznań, Poland ## Toeplitz operators on backward shift invariant subspaces of $H^p$

The talk is based on a joint work with Maria Nowak from Maria Curie-Skłodowska University in Lublin. We present extensions of some known results on compressed Toeplitz operators on the model spaces to the backward shift invariant subspaces of $H^p$, $1 \lt p \lt \infty$.

- Shailesh Trivedi

Indian Institute of Technology Kanpur, India ## Analytic $m$-isometries without the wandering subspace property

The wandering subspace problem for an analytic norm-increasing $m$-isometry $T$ on a Hilbert space $\mathcal H$ asks whether every $T$-invariant subspace of $\mathcal H$ can be generated by a wandering subspace. An affirmative solution to this problem for $m=1$ is ascribed to Beurling-Lax-Halmos, while that for $m=2$ is due to Richter. In this talk, we capitalize on the idea of weighted shift on one-circuit directed graph to construct a family of analytic cyclic $3$-isometries, which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one dimensional space, their norms can be made arbitrarily close to $1$. We also show that if the wandering subspace property fails for an analytic norm-increasing $m$-isometry, then it fails miserably in the sense that the smallest $T$-invariant subspace generated by the wandering subspace is of infinite codimension.

This is a joint work with Akash Anand and Sameer Chavan.