Abstracts

We show that dual truncated Toeplitz operators on the orthogonal complement of a model space are equivalent after extension to certain paired operators, and we use this  to study their kernels and their spectral properties.

Based on joint work with Kamila Kliś-Garlicka, Bartosz Łanucha and Marek Ptak.

Consider the standard weighted Fock spaces $F^p_{\alpha}$ on $\mathbb{C}^n$, where $p \in (1,\infty)$ and $\alpha > 0$. The theory of Fock spaces is in many regards very similar to the classical Bergman space theory. However, there are a few key differences, which make the theory interesting. One of them concerns Hankel operators, which are denoted by $H_f$ for bounded symbols $f$. In the Hilbert space case ($p = 2$) a well-known theorem by Berger and Coburn states that $H_f$ is compact if and only if $H_{\bar{f}}$ is compact. On the other hand, it is easily seen that this statement is wrong for Bergman spaces. Zhu comments: “A partial explanation for this difference is probably the lack of bounded analytic or harmonic functions on the entire complex plane.” Using limit operator theory, I will give a new proof of the Berger-Coburn result, which also includes the Banach space cases ($p \neq 2$) and fully explains this difference between Bergman and Fock spaces. Namely, it will be apparent that the only ingredient missing for the same proof to work on Bergman spaces is Liouville's theorem. Based on joint work with Jani Virtanen.

We study certain densely defined unbounded operators on the Segal-Bargmann space. These are the annihilation and creation operators of quantum mechanics. In several complex variables we have the $\partial$-operator and its adjoint $\partial^*$ acting on $(p,0)$-forms with coefficients in the Segal-Bargmann space. We consider the corresponding $\partial$-complex and study spectral properties of the corresponding complex Laplacian $\tilde \Box = \partial \partial^* + \partial^*\partial.$ Finally we study a more general complex Laplacian $\tilde \Box_D = D D^* + D^* D,$ where $D$ is a differential operator of polynomial type, to find the canonical solutions to the inhomogeneous equations $Du=\alpha$ and $D^*v=\beta.$

We also study the $\partial$-complex on several models including the complex hyperbolic space, which turns out to have duality properties similar to the Segal-Bargmann space (which is common work with Duong Ngoc Son).

We obtain complete characterizations in terms of Carleson measures for bounded/compact differences of weighted composition operators acting on the standard weighted Bergman spaces over the unit disk. Unlike the known results, we allow the weight functions to be non-holomorphic and unbounded.

As a consequence we obtain a compactness characterization for differences of unweighted composition operators acting on the Hardy spaces in terms of Carleson measures and, as a nontrivial application of this, we show that compact differences of composition operators with univalent symbols on the Hardy spaces are exactly the same as those on the weighted Bergman spaces. As another application, we show that an earlier characterization due to Acharyya and Wu for compact differences of weighted composition operators with bounded holomorphic weights does not extend to the case of non-holomorphic weights. We also include some explicit examples related to our results.

It is known that for Cowen-Douglas operators, the curvature of the corresponding eigenvector bundles play an important role in their classification up to unitary equivalence or similarity. However, the curvature is not so easy to calculate in general. We consider a subclass of the Cowen-Douglas class in which things are more tractable. This talk is based on joint work with K. Ji, J. Sarkar, and J. Xu.

In this talk, we consider the problem: when Möbius invariant function spaces are continuously embedded in tent spaces. It is also known as Carleson measure problem. We will introduce the recent development of Carleson measure for some well-known analytic function spaces, then present our work on Carleson measure for $\operatorname{BMOA}$, Bloch space and $Q_p$ spaces. (This is a joint work with K. Zhu)

In this talk, we study the Banach algebras $\mathcal{T}(\mathbb{T}_m^q)$ which is generated by Toeplitz operators whose symbols are invariant under the action $\mathbb{T}_m^q$ subgroup of the maximal torus $\mathbb{T}^n$, which are acting on the Bergman space on weakly pseudo-convex domains $ \Omega^n_p$. Moreover, we proved that the commutator of the $C*$ algebra $\mathcal{T}(\mathcal{R}_k(\Omega^n_p))$ is equal to The Toeplitz algebra $\mathcal{T}(\mathbb{T}_m^q)$, where $\mathcal{T}(\mathcal{R}_k(\Omega^n_p))$ is the $C^*$ algebra generated by Toeplitz operators where the symbols are $k$-radials. Finally, using this relationship we found some commutative Banach algebras generated by Toeplitz operators which generalized the Banach algebra generated by Toeplitz operators with quasi-homogeneous quasi-radial symbols.

This is a joint work with Mauricio Hernández-Marroquin and Luis Alfredo Dupont-García.

Determinantal varieties of images of Coxeter generators were shown to determine representations of non-exceptional finite Weyl groups up to unitary equivalence by Cuckovic, Stessin, and Tchernev. The main result established shows determinantal varieties of a larger set (more than the generating set) of elements determine the character of representations of affine Weyl groups $\tilde{B}_n$, $\tilde{C}_n$, and $\tilde{D}_n$.

Carleson measures are fundamental to the study of holomorphic function spaces, as they are connected to the characterization of the corresponding multiplier algebras, existence of boundary values, interpolating sequences, Hankel-type operators, Corona problems, etc. I will discuss a description of the Carleson measures for the Dirichlet space of the bidisc, in terms of a newly developed bi-parameter potential theory which is based on kernels of tensor-product structure. There are very significant differences to classical potential theory. In particular, the maximum principle fails. Yet, perhaps surprisingly, the bi-parameter theory completely characterizes the Carleson measures, in analogy with Stegenga’s description of the Carleson measures for the Dirichlet space of the disc.

Based on joint work with Nicola Arcozzi, Pavel Mozolyako, and Giulia Sarfatti.

Let $D = G/K \subset \mathbb{C}^n$ be an irreducible bounded symmetric domain circled around the origin, where $G$ is a reductive group with an action on $D$ that realizes the biholomorphism group of $D$ with $K$ the isotropy subgroup at the origin. For any closed subgroup $H$ of $K$ let $\mathcal{A}^H$ be the essentially bounded measurable symbols on $D$ that are $H$-invariant, and let us denote by $\mathcal{T}^{(\lambda)}(\mathcal{A}^H)$ the $C^*$-algebra generated by the Toeplitz operators with symbols in $\mathcal{A}^H$ acting on the weighted Bergman space $\mathcal{H}^2_\lambda(D)$. It is well known that $\mathcal{T}^{(\lambda)}(\mathcal{A}^K)$ is commutative for $D$ as above. But $\mathcal{T}^{(\lambda)}(\mathcal{A}^T)$ is commutative for $T$ a maximal torus in $K$ if and only if $D$ is biholomorphic to a unit ball. A question posed by Nikolai Vasilevski is to find out whether, for $D$ not biholomorphic to a unit ball, there exists $H$ a closed proper subgroup of $K$ containing a maximal torus such that $\mathcal{T}^{(\lambda)}(\mathcal{A}^H)$ is commutative.

In this talk we will show that there is such a subgroup that gives commutative $C^*$-algebras, as required, in the case of the classical domain $D^I_{2,2}$ consisting of complex $2\times 2$ matrices $Z$ such that $Z Z^* < I_2$. The biholomorphism group for this domain can be realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$ which contains the maximal torus $\mathbb{T}^2\times\mathbb{T}^2$ given by pairs of diagonal unitary $2\times 2$ matrices. For this setup, we will prove that for the group $H = \mathrm{U}(2)\times\mathbb{T}^2$ the $C^*$-algebra $\mathcal{T}^{(\lambda)}(\mathcal{A}^H)$ is commutative. The proof is based in representation theory and allows us to provide a description of the spectra of the corresponding Toeplitz operators.

This is joint work with Gestur Olafsson and Matthew Dawson.

Let $H$ be a RKHS of functions regarded as a subspace of $L^2(G\times Y)$, where $G$ is a locally compact abelian group provided with a Haar measure and $Y$ is a measure space. Suppose $H$ to be invariant under "horizontal translations" naturally associated to $G$. Under some technical assumptions, we study the W*-algebra $\mathcal{V}$ of all translation-invariant bounded linear operators acting on $H$. For this purpose, we apply the operator $F \otimes I$, i.e. the Fourier transform with respect to the first coordinate, and decompose the image $\widehat{H}$ of $H$ into the direct integral of the fibers $\widehat{H}_{\xi}$, where $\xi\in\widehat{G}$. We conclude that $\mathcal{V}$ is commutative if and only if all fibers have dimension $0$ or $1$. If this happens, we construct a unitary operator that simultaneously diagonalizes all operators in $\mathcal{V}$. Our scheme generalizes several results previously found by Vasilevski, Quiroga-Barranco, Grudsky, Karapetyants, Hutník, Loaiza, Lozano, Sánchez-Nungaray, Ramírez Ortega, Esmeral, and other authors.

As a new example, we describe the W*-algebra of "vertical" operators in the polyharmonic space over the upper half-plane.

This talk is based on a joint work with Crispin Herrera Yañez and Egor Maximenko.

In this talk we give decompositions of the W*-algebras of radial operators on the spaces $L^2(\mathbb{C},d\mu_G)$, $F_n$ and $F_{(n)}$, where $d\mu_g=\frac{1}{\pi}e^{-\vert z\vert^2}$ is the gaussian weight on the complex plane, $F_n$ is the $n$-th polyanalytic Bargmann-Segal-Fock space and $F_{(n)}=F_n\ominus F_{n-1}$ is the $n$-th true polyanalytic Fock space.

We construct an orthonormal basis for $L^2(\mathbb{C},d\mu_G)$ using creation operators, and show its equivalence to complex Hermit polynomials, denoted $(b_{p,q})_{p,q=0}^\infty$. Using ideas from Vasilevski (2000), we prove explicit formulas for the reproducing kernels of $F_{(n)}$ and $F_n$: $$K_z^{(n)}(w)=e^{\overline{z}w}L_{n-1}(\vert w-z\vert^2), \qquad\qquad K_z^{n}(w)=e^{\overline{z}w}L^1_{n-1}(\vert w-z\vert^2),$$ where $L^\alpha_n$ is the $n$-th generalized Laguerre polynomial of degree $\alpha$.

Let $\mathcal{D}_d$ be the $d$th diagonal subspace of $L^2(\mathbb{C},d\mu_G)$ defined as the closed\\subspace generated by $b_{j,k}$ with $j-k=d$. We prove that these diagonal

1. the W*-algebra of radial operators on $L^2(\mathbb{C},d\mu_G)$ decomposes into the direct sum $\bigoplus_{d\in\mathbb{Z}} B(\mathcal{D}_d)$,
2. the radial operators on $F_{(n)}$ are diagonal with respect to the orthonormal basis $(b_{p,n-1})_{p=0}^\infty$,
3. the W*-algebra $\mathcal{R}_n$ of radial operators on $F_n$ is isomorphic to the following W*-algebra of sequences of square matrices:

I am going to speak about recent work with Dan Timotin and Mohamed Zarrabi showing in which ways the classical Szego theorem about eigenvalues of Toeplitz matrices can be generalized to truncated Toeplitz operators.

We review a recent approach to weighted Bergman spaces $A_v^p$, $1 \lt p \lt \infty$, or $H_v^\infty$ on the unit disc and also related spaces of entire  functions: we use techniques where the Taylor series of the analytic functions are divided into an infinite number of blocks, which consist of polynomials with given fixed degrees somehow related to the given weight of the Bergman norm. This allows to write an expression of the weighted Bergman norm, which is useful and applicable, if  $p \not=2$ (in the case $p=2$ our results do not yield new information). We apply the techniques to describe solid hulls and cores of the spaces, and characterize the boundedness and compactness of some sequence space multipliers. We also give a characterization  of the boundedness and compactness of Toeplitz operators with radial symbols in the space $H_v^\infty$ on the disc. The work is in co-operation with José Bonet (Valencia) and Wolfgang Lusky (Paderborn).