One computes the joint and essential joint spectra of a pair of multiplication operators with bounded analytic functions on the Hardy spaces of the unit ball in ${ℂ}^n$.

The purpose of this paper is to study the Sarason’s problem on Fock spaces of polyanalytic functions. Namely, given two polyanalytic symbols $f$ and $g$, we establish a necessary and sufficient condition for the boundedness of some Toeplitz products $T_f{T}_{\overline{g}}$ subjected to certain restriction on $f$ and $g$. We also characterize this property in terms of the Berezin transform.

In 1997 Pták defined generalized Hankel operators as follows: Given two contractions $T_1\in ℬ\left({\mathscr{H}}_1\right)$ and $T_2\in ℬ\left({\mathscr{H}}_2\right)$, an operator $X{\mathscr{H}}_1\to {\mathscr{H}}_2$ is said to be a generalized Hankel operator if $T_{2}X=X{T}_1^{*}$ and $X$ satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of $T_1$ and $T_2$. This approach, call it (P), contrasts with a previous one developed by Pták and Vrbová in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat...

The paper discusses Problems 8 and 88 posed by Stanisław Mazur in the Scottish Book. It turns out that negative solutions to both problems are immediate consequences of the results of Peller [J. Operator Theory 7 (1982)]. We discuss here some quantitative aspects of Problems 8 and 88 and give answers to open problems discussed in a recent paper of Pełczyński and Sukochev in connection with Problem 88.

We prove that the self-commutator of a Toeplitz operator with unbounded analytic rational symbol has a dense domain in both the Bergman space and the Hardy space of the unit disc. This is a basic step towards establishing whether the self-commutator has a compact or trace-class extension.

For any holomorphic function $f$ on the unit polydisk ${𝔻}^n$ we consider its restriction to the diagonal, i.e., the function in the unit disc $𝔻\subset ℂ$ defined by $\mathrm{Diag}f\left(z\right)=f(z,...,z)$, and prove that the diagonal map $\mathrm{Diag}$ maps the space $Q_{p,q,s}\left({𝔻}^n\right)$ of the polydisk onto the space ${\widehat{Q}}_{p,s,n}^q\left(𝔻\right)$ of the unit disk.

If G is a discrete group, the algebra CD(G) of convolution dominated operators on l²(G) (see Definition 1 below) is canonically isomorphic to a twisted L¹-algebra $l¹(G,l^{\infty }\left(G\right),T)$. For amenable and rigidly symmetric G we use this to show that any element of this algebra is invertible in the algebra itself if and only if it is invertible as a bounded operator on l²(G), i.e. CD(G) is spectral in the algebra of all bounded operators. For G commutative, this result is known (see [1], [6]), for G noncommutative discrete...

One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.

We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed...

We present sufficient conditions for the existence of $p$th powers of a quasihomogeneous Toeplitz operator $T_{{\mathrm{e}}^{\mathrm{i}s\theta }\psi }$, where $\psi$ is a radial polynomial function and $p$, $s$ are natural numbers. A large class of examples is provided to illustrate our results. To our best knowledge those examples are not covered by the current literature. The main tools in the proof of our results are the Mellin transform and some classical theorems of complex analysis.