We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

We study the $L^p$-boundedness of the Berezin transform on the generalised Hartogs triangles which are defined by

$$ \begin{align*}H_k:=\{(z, w)\in\mathbb C^n\times\mathbb C: |z_1|^2+\cdots+|z_n|^2<|w|^{2k}<1\},\end{align*} $$

where $z=(z_1, \ldots , z_n)$ and $k\in \mathbb N$. We prove that the Berezin transform is bounded on $L^p(H_k)$ if and only if $p>nk+1$.

Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$, we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\varphi $, respectively, on the weighted Bergman space $A^2_\alpha $. The ranges of $D_\varphi $ and $D_{\overline \varphi }$, written as $H(\varphi )$ and $H(\overline \varphi )$ and equipped with appropriate inner products, are called sub-Bergman spaces.

We prove the following three results in the paper: for $-1<\alpha \le 0$, the space $H(\varphi )$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius map; for $\alpha>-1$, we have $H(\varphi )=H(\overline \varphi )=A^2_{\alpha -1}$ if and only if the defect operators $D_\varphi $ and $D_{\overline \varphi }$ are compact; and for $\alpha>-1$, we have $D^2_\varphi (A^2_\alpha )= D^2_{\overline \varphi }(A^2_\alpha )=A^2_{\alpha -2}$ if and only if $\varphi $ is a finite Blaschke product. In some sense, our restrictions on $\alpha $ here are best possible.

We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$, (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant $V$, our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.

We characterize the membership in the Schatten ideals $\mathcal {S}_p$, $0<p<\infty $, of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given.

Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

Let $\Omega $ be a bounded Reinhardt domain in $\mathbb {C}^n$ and $\phi _1,\ldots ,\phi _m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi _m}\cdots T_{\phi _1}=0$ on the Bergman space on $\Omega $ , then $\phi _j=0$ for some j.

This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.

We characterize all bounded Hankel operators $\unicode[STIX]{x1D6E4}$ such that $\unicode[STIX]{x1D6E4}^{\ast }\unicode[STIX]{x1D6E4}$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.

We find the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space Ap,α

\begin{equation*} \|H\|_{A^{p,\alpha}\rightarrow A^{p,\alpha}}\geq\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}, \,\, \textnormal{for} \,\, 1<\alpha+2<p. \end{equation*}

$\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}$

We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari and Fefferman are proved.

Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

For the Bergman projection operator $P$ we prove that

$$\left\| P:\,{{L}^{1}}\left( B,\,d\lambda \right)\,\to \,{{B}_{1}} \right\|\,=\,\frac{\left( 2n\,+\,1 \right)!}{n!}.$$

Here $\lambda$ stands for the hyperbolic metric in the unit ball $B$ of ${{\mathbb{C}}^{n}}$, and ${{B}_{1}}$ denotes the Besov space with an adequate semi-norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.

Let ${{H}^{2}}\left( \Omega \right)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \,\subset \,{{\mathbb{C}}^{n}}$, and let $A\,\subset \,{{L}^{\infty }}\left( \partial \Omega \right)$ denote the subalgebra of all ${{L}^{\infty }}$-functions $f$ with compact Hankel operator ${{H}_{f}}$. Given any closed subalgebra $B\,\subset \,A$ containing $C\left( \partial \Omega \right)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal{T}\left( B \right)\,\subset \,B\left( {{H}^{2}}\left( \Omega \right) \right)$. In particular, we show that every derivation on $\mathcal{T}\left( A \right)$ is inner. These results are new even for $n\,=\,1$, where it follows that every derivation on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C \right)$ is inner, while there are non-inner derivations on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C\left( \partial {{\mathbb{B}}_{n}} \right) \right)$ over the unit ball ${{\mathbb{B}}_{n}}$ in dimension $n\,>\,1$.

In this short note, we study a relation between the tensor product of matrices and a multilinear map defined by the optimal operator. In this particular case, the linear transform (mediating morphism) hidden in the abstract definition of the general tensor product can be determined explicitly.

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.