Let $\{R_n\}$ and $\{P_n\}$ be two polynomial systems which induce signed polynomial hypergroup structures on $\mathbb{N}_0.$ We investigate when the Banach algebra $l^1(\mathbb{N}_0,h^R)$ can be continuously embedded into or is isomorphic to $l^1(\mathbb{N}_0,h^P).$ We find sufficient conditions on the connection coefficients $ c_{nk} $ given by $ R_n = \sum_{k=0}^n c_{nk} P_k, $ for the existence of such an embedding or isomorphism. Finally we apply these results to obtain amenability-properties of the $l^1$-algebras induced by Bernstein-Szego and Jacobi polynomials.

In this paper we study a possibility of a decomposition of a bounded operator in a Hilbert space $H$ as a product of a J-unitary and a J-self-adjoint operators, where J is a conjugation (an antilinear involution). This decomposition shows an inner structure of a bounded operator in a Hilbert space. Some decompositions of J-unitary and unitary operators which generalize decompositions in the finite-dimensional case are also obtained. Matrix representations for J-symmetric and J-skew-symmetric operators are studied. Simple basic properties of J-symmetric, J-skew-symmetric and J-isometric operators are obtained.

This article presents an interaction between functional homomorphisms, dynamical systems and differential equations. The exponential functions play pivotal role in this interaction.

It is known that $C(X)$ is algebraically closed if $X$ is a locally connected, hereditarily unicoherent compact Hausdorff space. For such spaces, we prove that if $F:C(X) \to C(X)$ is an entire function in the sense of Lorch, i.e., is given by an everywhere convergent power series with coefficients in $C(X)$, and satisfies certain restrictions, then it has a root in $C(X)$. Our results generalizes the monic algebraic case.

The Fubini theorem for the generalized Dobrakov integral in complete bornological locally convex topological vector spaces is proven.

We show that there is an affine order isomorphism between completely positive maps from a C*-algebra $A$ to the C*-algebra $L(H)$ of all bounded linear operators on a Hilbert space $H,$ $u$-covariant with respect to a C*-dynamical system $\left( G,\alpha ,A\right)$ and $u$-covariant completely positive maps from the crossed product $A\times _{\alpha }G$ to $L(H)$, which preserves the Lebesgue decomposition.

The well-known Mathieu series

\begin{eqnarray*} S_M(r) = \sum_{n=1}^{\infty}\frac{2n}{(n^2+r^2)^2} \qquad (r>0), \end{eqnarray*}

can be transformed into the form

\begin{eqnarray*} S_M(r) = \frac{1}{2r}\sum_{n=1}^{\infty} \frac{\sqrt{nr}^2}{\left(\sqrt{\frac{n^2+r^2}{2}}\right)^4} =\frac{1}{2r}\sum_{n=1}^{\infty}\frac{G^2(n,r)}{Q^4(n,r)}, \end{eqnarray*}

where $G(n,r)$ and $Q(n,r)$ denote the Geometric and Quadratic mean of $n\in \mathbb N$ and $r>0$. This connection leads us to the idea to introduce and research the so--called Quotient mean series as a be a generalizations of Mathieu's and Mathieu--type series. We give an integral representation of such series and their alternating variant, together with associated inequalities. Also, special cases of quotient mean series, involving Bessel function of the first kind, have been studied in detail.

In this paper, by using the methods of real analysis and functional analysis, a Hilbert-type integral inequality in the subinterval $(a,\infty )\,\,(a>0)$ with the homogeneous kernel of $-\lambda $-degree and a best constant factor and its operator expression are given. As applications, a few improved results, the equivalent forms and some new inequalities with the particular kernels are obtained.

Sufficient conditions for a topological *-algebra under which several Hermitian inner products are mutually equivalent are given.

Let $A$ be an algebra. A sequence $\{d_n\}$ of linear mappings on $A$ is called a higher derivation if $d_n(ab)=\sum_{k=0}^n d_k(a)d_{n-k}(b)$ for each $a,b\in A$ and each nonnegative integer $n$. In this paper a notion of an inner higher derivation is given. We characterize all uniformly bounded inner higher derivations on Banach algebras and show that each uniformly bounded higher derivation on a Banach algebra $A$ is inner provided that each derivation on $A$ is inner.

We solve classical differential equations, linear and affine, in a more general setting than usual; no use is made of manifolds. We work within the context of general $A$-convex algebras.

Using the concept of exponential monomial on Sturm--Liouville hypergroups we show that an important subclass of exponential monomials, the class of special exponential monomials has a linear independence property. The result can be reformulated as the linear independence of the derivatives with respect to the parameter of the solutions of eigenvalue problems for second order linear differential equations.

In this paper we obtain a characterization of finite-dimensional Hilbert C*-modules. It is known that those are the modules for which both underlying C*-algebras are finite-dimensional. We show that such modules can be described by a certain property of bounded sequences of their elements. It turns out that similar property leads to another characterization of Hilbert C*-modules over C*-algebras of compact operators.

In this paper the conditional multipliers acting between $L^p$ spaces are characterized by using some properties of conditional expectation operator. Also, we determine the essential norm of $uC_\varphi$ on $L^p$ for $p \in (1,\infty)$.

The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix polynomials are underlined.