The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). Our goal is to derive their basic properties including the orthogonality properties and Rodrigues formula. Furthermore, we define a new polynomial associated with the Hermite-Hermite matrix polynomials and establish the matrix differential equation associated with these polynomials. We give the addition theorems, multiplication theorems and summation formula for the Hermite-Hermite...

This article is a survey of some Tauberian theorems obtained recently in connection with work on asymptotic behaviour of semigroups of operators on Banach spaces. The results in operator theory are described in [6], where we made little attempt to show the Tauberian aspects. At the end of this article, we will give a sketch of the connections between the results in this article and in [6]; for details, the reader can turn to the original papers. In this article, we make no attempt to describe...

The paper is concerned with the spectral analysis for the class of linear operators $A=D_{\lambda }+X\otimes Y$ in non-archimedean Hilbert space, where $D_{\lambda }$ is a diagonal operator and $X\otimes Y$ is a rank one operator. The results of this paper turn out to be a generalization of those results obtained by Diarra.

We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au\left(x\right)=\left({\sum }_{k=1}^N{e}^{{\phi }_k}{T}_{{\alpha }_k}\right)u\left(x\right)={\sum }_{k=1}^N{e}^{{\phi }_k\left(x\right)}u\left({\alpha }_k\left(x\right)\right)$, u ∈ C(X), where ${\phi }_k\in C\left(X\right)$ and ${\alpha }_k:X\to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $\phi ={\left({\phi }_k\right)}_{k=1}^N$. We prove that $ln\left(r\left(A\right)\right)=\lambda \left(\phi \right)=ma{x}_{\nu \in Mes}{\sum }_{k=1}^N{\int }_X{\phi }_{k}d{\nu }_k-\lambda *\left(\nu \right)$, where Mes is the set of all probability vectors of measures $\nu ={\left({\nu }_k\right)}_{k=1}^N$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on $\left(C^N\left(X\right)\right)*$. In other...

We prove that for the spectral radius of a weighted composition operator $a{T}_{\alpha }$, acting in the space $L^p(X,,\mu )$, the following variational principle holds: $lnr\left(a{T}_{\alpha }\right)=ma{x}_{\nu \in M{¹}_{\alpha ,e}}{\int }_{X}ln\left|a\right|d\nu$, where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, $M{¹}_{\alpha ,e}$ is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and ${}_{\infty }$-measurable function, where ${}_{\infty }={\bigcap }_{n=0}^{\infty }{\alpha }^{-n}\left(\right)$. This considerably extends the range of validity of the above formula, which was previously known in the case...

We define on an ordered semi simple symmetric space $ℳ=G/H$ a family of spherical functions by an integral formula similar to the Harish-Chandra integral formula for spherical functions on a Riemannian symmetric space of non compact type. Associated with these spherical functions we define a spherical Laplace transform. This transform carries the composition product of invariant causal kernels onto the ordinary product. We invert this transform when $G$ is a complex group, $H$ a real form of $G$, and when $ℳ$...

The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces, subspaces of the space of rapidly decreasing smooth complex functions, which are defined by imposing suitable bounds on their elements in terms of a given sequence M. Necessary and sufficient conditions on M are stated for the problem to have a solution, sometimes coming with linear continuous right inverses of the moment map, sending a function to the sequence of its moments. On the way, some results on the...

An abelian $*$-semigroup $S$ is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on $S$ admits a unique disintegration as an integral of hermitian multiplicative functions (resp. nonnegative such). We prove that every Stieltjes perfect semigroup is perfect. The converse has been known for semigroups with neutral element, but is here shown to be not true in general. We prove that an abelian $*$-semigroup $S$ is perfect if for each $s\in S$ there exist $t\in S$ and $m,n\in {\mathbb{N}}_0$ such that $m+n\ge 2$...