We study Carleson measures and Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip’s results from the unit disk of $ℂ$ to the unit ball of ${ℂ}^n$. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.

This paper characterizes the commutant of certain multiplication operators on Hilbert spaces of analytic functions. Let $A=M_z$ be the operator of multiplication by z on the underlying Hilbert space. We give sufficient conditions for an operator essentially commuting with A and commuting with $A^n$ for some n>1 to be the operator of multiplication by an analytic symbol. This extends a result of Shields and Wallen.

We prove the Schatten-Lorentz ideal criteria for commutators of multiplications and projections based on the Calderón reproducing formula and the decomposition theorem for the space of symbols corresponding to commutators in the Schatten ideal.

We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.

We show that a bounded linear operator S on the weighted Bergman space A¹(ψ) is compact and the predual space A₀(φ) of A¹(ψ) is invariant under S* if and only if $S{k}_z\to 0$ as z → ∂D, where $k_z$ is the normalized reproducing kernel of A¹(ψ). As an application, we give conditions for an operator in the Toeplitz algebra to be compact.

We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on the weighted Bergman space $A^2\left(\Omega \right)$, where $\Omega$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^p{\overline{z}}^q}$ to be complex symmetric. When $p,q\in {\mathbb{N}}^2$, we prove that $T_{z^p{\overline{z}}^q}$ is complex symmetric on $A^2\left(\Omega \right)$ if and only if $p_1=q_2$ and $p_2=q_1$. Moreover, we completely characterize when monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on $A^2\left({𝔻}_n\right)$ are $J_U$-symmetric with the $n\times n$ symmetric unitary matrix $U$.

The Hilbert matrix acts on Hardy spaces by multiplication with Taylor coefficients. We find an upper bound for the norm of the induced operator.

This paper studies the compression of a $k$th-order slant Toeplitz operator on the Hardy space $H^2\left({𝕋}^n\right)$ for integers $k\ge 2$ and $n\ge 1$. It also provides a characterization of the compression of a $k$th-order slant Toeplitz operator on $H^2\left({𝕋}^n\right)$. Finally, the paper highlights certain properties, namely isometry, eigenvalues, eigenvectors, spectrum and spectral radius of the compression of $k$th-order slant Toeplitz operator on the Hardy space $H^2\left({𝕋}^n\right)$ of $n$-dimensional torus ${𝕋}^n$.

For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [fgl08a, fgl08]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the...