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We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of…

Let $A_{\alpha}^{p}(\mathbb{B}^n;\mathbb{C}^d)$ be the weighted Bergman space on the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ of functions taking values in $\mathbb{C}^d$. For $1<p<\infty$ let $\mathcal{T}_{p,\alpha}$ be the algebra…

When is the collection of $\mathsf S$-Toeplitz operators with respect to a tuple of commuting bounded operators $\mathsf S= (S_1, S_2, \ldots , S_{d-1}, P)$, which has the symmetrized polydisc as a spectral set, non-trivial? The answer is…

We revisit the theory of operator-valued free convolution powers given by a completely positive map $\eta$. We first give a general result, with a new analytic proof, that the $\eta$-convolution power of the law of $X$ is realized by…

Quantization and spectral properties of Toeplitz operators acting on spaces of pluriharmonic functions over bounded symmetric domains and $\mathbb C^n$ are discussed. Results are presented on the asymptotics \begin{align*} \|…

The Dirac-Dolbeault operator for a compact K\"ahler manifold is a special case of a Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows to express the values of the sections of the…

We derive a criterion that is convenient for applications and exactly characterizes the test function space on which the operator realization of a given series of Wick powers of a free field is possible. The suggested derivation does not…

We study the spectra and pseudospectra of finite and infinite tridiagonal random matrices, in the case where each of the diagonals varies over a separate compact set, say $U,V,W\subset\mathbb{C}$. Such matrices are sometimes termed…

In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables. We show that Wolf's theorem is a special case of a general…

We consider Toeplitz operators in Bergman and Fock type spaces of polyanalytic $L^2\textup{-}$functions on the disk or on the half-plane with respect to the Lebesgue measure (resp., on $\mathbb{C}$ with the plane Gaussian measure). The…

In this article, we show that if $A$ is a maximal monotone operator on a Hilbert space $H$ with $0$ in the range $\textrm{Rg}(A)$ of $A$, then for every $0<s<1$, the Dirichlet problem associated with the Bessel-type equation $$…

We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and R\'ev\'esz (2024). Our method does not only apply to the Riemann zeta…

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the…

This paper mainly studies totally Abelian operators in the context of analytic Toeplitz operators on both the Hardy and Bergman space. When the symbol is a meromorphic function on $\mathbb{C}$, we establish the connection between totally…

We consider the Toeplitz operators on the weighted Bergman spaces over the unit ball $\mathbb{B}^n$ and their analytic continuation. We proved the commutativity of the $C^*-$algebras generated by the analytic continuation of Toeplitz…

We consider the family of Toeplitz operators $T_{J\bar S^{a}}$ acting in the Hardy space $H^2$ in the upper halfplane; $J$ and $S$ are given meromorphic inner functions, and $a$ is a real parameter. In the case where the argument of $S$ has…

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit…

We study Bergman spaces A^2(D), their kernels and Toeplitz operators on unbounded, doubly periodic domains D in the complex plane. We establish the mapping properties of the Floquet transform operator defined in A^2(D) and derive a general…

We look at Toeplitz operators $T_\nu$ on the Fock Space (also known as the Segal-Bargmann space) which have a positive Borel measure $\nu$ as a symbol. We characterize when $\left(T_\nu\right)^s$ for $0<s\leq 1$ is in the symmetrically…

The definition of Toeplitz operators in the Bergman space $A^2(D)$ of square integrable analytic functions in the unit disk in the complex plane is extended in such way that it covers many cases where the traditional definition does not…