2000 Mathematics Subject Classification: Primary: 47B47, 47B10; secondary 47A30.Let H be an infinite-dimensional complex Hilbert space and let A, B ∈ L(H), where L(H) is the algebra of operators on H into itself. Let δAB: L(H) → L(H) denote the generalized derivation δAB(X) = AX − XB. This note will initiate a study on the class of pairs (A,B) such that [‾(R(δAB))] = [‾(R(δB*A*))]; i.e. [‾(R(δAB))] is self-adjoint.

The paper relates several generalized eigenfunction expansions to classical spectral decomposition properties. From this perspective one explains some recent results concerning the classes of decomposable and generalized scalar operators. In particular a universal dilation theory and two different functional models for related classes of operators are presented.

In this paper we deal with weakly upper semi-continuous set-valued maps, taking arbitrary non-empty values, from a non-metric domain to a Banach space. We obtain selectors having the point of continuity property relative to the norm topology for a large class of compact spaces as a domain. Exact conditions under which the selector is of the first Borel class are also investigated.

By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane C which are square integrable with respect to a weight of the type e-Q(z), where Q(z) is a quadratic form such that tr Q > 0. Each such space is in a natural way associated with an (oriented) circle C in C. We consider the problem of interpolation between two Fock spaces. If C0 and C1 are the corresponding circles, one is led to consider the pencil of circles generated by C0 and C1....

The convexity and compactness in the weak operator topology of the image and pre-image of a generalized fractional linear transformation is established. As an application the exponential dichotomy of solutions to evolution problems of the parabolic type is proved.

Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E{*}_{\omega *}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G\left(x\right):={\int }_{\Omega }g(s,x\left(s\right))d\mu \left(s\right)$. Consider the integral functional G defined on some non-$L^p$-type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued C-subgradient)...

Let $𝔄=\left(\begin{array}{cc}𝒜& ℳ\\ & ℬ\end{array}\right)$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $(𝒜,ℬ)$-bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $[𝔏,𝔄]\subseteq 𝔏$. A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$, every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$.

Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator ${}_{a,b}$ by ${}_{a,b}\left(f\right)\left(z\right)=\Gamma (a+1)/\Gamma (b+1){\int }_0^1\left(f\left(t\right){(1-t)}^b\right)/\left({(1-tz)}^{a+1}\right)dt$, where a and b are non-negative real numbers. In particular, for a = b = β, ${}_{a,b}$ becomes the generalized Hilbert operator ${}_{\beta }$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that ${}_{a,b}$ is bounded on Dirichlet-type spaces $S^p$, 0 < p < 2, and on Bergman spaces $A^p$, 2 < p < ∞. Also we find an upper bound for the norm of the operator ${}_{a,b}$. These generalize...