We introduce the notion of Engliš algebras, defined in terms of reproducing kernels and Berezin symbols. Such algebras were apparently first investigated by Engliš (1995). Here we give some new results on Engliš C*-algebras on abstract reproducing kernel Hilbert spaces and some applications to various questions of operator theory. In particular, we give applications to Riccati operator equations, zero Toeplitz products, and the existence of invariant subspaces for some operators.

Let $\mu$ be a finite positive measure on the unit disk and let $j\ge 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{\mu }^{\left(j\right)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$ on the Bergman space $A^2$, and then give a sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class $(0<p<1)$ on $A^2$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^{\infty }(D,\mathrm{d}A)$ and $1<p<\infty$, we define the generalized Toeplitz...

A full description of the membership in the Schatten ideal $S_p\left(A{²}_{\omega }\right)$ for 0 < p < ∞ of Toeplitz operators acting on large weighted Bergman spaces is obtained.

Let $B_w\left({\ell }^p\right)$ denote the space of infinite matrices $A$ for which $A\left(x\right)\in {\ell }^p$ for all $x={\left\{x_k\right\}}_{k=1}^{\infty }\in {\ell }^p$ with $|x_k|\searrow 0$. We characterize the upper triangular positive matrices from $B_w\left({\ell }^p\right)$, $1<p<\infty$, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.

We consider separately radial (with corresponding group ${𝕋}^n$) and radial (with corresponding group $\mathrm{U}\left(n\right))$ symbols on the projective space ${\mathbb{P}}^n\left(ℂ\right)$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^{*}$-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the...

Under regularity assumptions, we establish a sharp large deviation principle for Hermitian quadratic forms of stationary Gaussian processes. Our result is similar to the well-known Bahadur-Rao theorem [2] on the sample mean. We also provide several examples of application such as the sharp large deviation properties of the Neyman-Pearson likelihood ratio test, of the sum of squares, of the Yule-Walker estimator of the parameter of a stable autoregressive Gaussian process, and finally of the empirical...

In 1968, Gohberg and Krupnik found a Fredholm criterion for singular integral operators of the form aP + bQ, where a,b are piecewise continuous functions and P,Q are complementary projections associated to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves. We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves.

In this paper the notion of slant Hankel operator $K_{\varphi }$, with symbol $\varphi$ in $L^{\infty }$, on the space $L^2\left(𝕋\right)$, $𝕋$ being the unit circle, is introduced. The matrix of the slant Hankel operator with respect to the usual basis $\{z^i:i\in \mathbb{Z}\}$ of the space $L^2$ is given by $\langle {\alpha }_{ij}\rangle =\langle a_{-2i-j}\rangle$, where $\sum _{i=-\infty }^{\infty }{a}_i{z}^i$ is the Fourier expansion of $\varphi$. Some algebraic properties such as the norm, compactness of the operator $K_{\varphi }$ are discussed. Along with the algebraic properties some spectral properties of such operators are discussed. Precisely, it is proved that for an invertible...

By a straightforward computation we obtain eigenvalue estimates for Toeplitz operators related to the two standard reproducing formulas of the wavelet theory. Our result extends the estimates for Calderón-Toeplitz operators obtained by Rochberg in [R2]. In the first section we recall two standard reproducing formulas of the wavelet theory, we define Toeplitz operators and discuss some of their properties. The second section contains precise statements of our results and their proofs. At the end...

We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space.