In this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$ is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition. We define the It^{o} stochastic integral of operator set-valued stochastic pr...

A Banach space E can be embedded into B(H) in a variety of ways. The embeddings giving rise to the minimal and maximal possible operator space norms (introduced in [1], [3] and [4] and further investigated in [21] and [22]) are especially interesting to us. Let A be the set of all functionals on E of norm not exceeding 1 and let B be the set of all linear maps T : E → Mn such that ||T || ≤ 1 (h...

Let φ and ψ be holomorphic maps on such that φ( ) ⊂ . Let Cφ,Mψ and D be the composition, multiplication and differentiation operators, respectively. In this paper, we consider linear operators induced by products of these operators from Bergman-Nevanlinna spaces AβN to Bloch-type spaces. In fact, we prove that these operators map AβN compactly into Bloch-type spaces if and only if they map A β...

The purpose of this paper is threefold: first to mainly review several recent results concerning the fine spectrum of the operator ∆v over the sequence spaces c and lp, where 1 < p < ∞; second to provide some new results concerning the residual spectrum and the continuous spectrum of the operator ∆v over the sequence spaces c and lp; and third to modify the definition of the operator ∆v and to ...

Several spaces defined according to different formulations for operator-valued functions in BMO are defined and studied.

This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A−), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum s...

We introduce the notion of Krein-operator convexity in the setting of Krein spaces. We present an indefinite version of the Jensen operator inequality on Krein spaces by showing that if (H , J) is a Krein space, U is an open set which is symmetric with respect to the real axis such that U ∩ R consists of a segment of real axis and f is a Kreinoperator convex function on U with f(0) = 0, then f(...

Recently, OH-type and OH-cotype of operator spaces, an operator space version of type and cotype, were introduced and investigated by the author. In this paper we define weak OH-type 2 (resp. weak OH-cotype 2) of operator spaces, which lies strictly between OH-type 2 (resp. OH-cotype 2) and OH-type p for all 1 ≤ p < 2. (resp. OH-cotype q for all 2 < q ≤ ∞) This is an analogue of weak type 2 and...