We give a representation for entanglement-breaking channels in separable Hilbert space that generalizes the “Kraus decomposition with rank one operators”and use it to describe the complementary channels. We also give necessary and sufficient condition of entanglement-breaking for a general quantum Gaussian channel. Application of this condition to onemode channels provides several new cases whe...

Let H be a separable complex Hilbert space. Denote by G∞(H) the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every continuous order preserving bijective transformation of G∞(H) is induced by an invertible bounded semi-linear operator.

If N ⊆M is an inclusion of type II1 factors of finite index on a separable Hilbert space, and if N has a Cartan subalgebra then we show that H(N ,M) = 0 for n ≥ 1. We also show that H cb(N ,M) = 0, n ≥ 1, for an arbitrary finite index inclusion N ⊆M of von Neumann algebras.

In this paper we describe the form of those continuous multiplicative maps on B(H) (H being a separable complex Hilbert space of dimension not less than 3) which preserve the rank, or the corank. Furthermore, we characterize those continuous *-semigroup endomorphisms of B(H) which are spectrum non-increasing.

In the present study, a fractional order differential equation with deviating argument is considered in a separable Hilbert space H . We will prove the existence and convergence of an approximate solution for the given problem by using the analytic semigroup theory and the fixed point method. Finally, we consider the Faedo-Galerkin approximation of the solution and prove some convergence results.

Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that its form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary times in the L-norm, that is, one has

Let G be a countable group of unitary operators on a complex separable Hilbert space H. We give a characterization of biorthogonality among Riesz multiwavelets in terms of certain invariant properties of their associated core spaces. A large family of non-biorthogonal Riesz multiwavelets is exhibited. We also discuss some results on linear perturbation of orthonormal multiwavelets.

We establish several conditions which are equivalent to |[Bx, x]| ≤ 〈Ax, x〉 , ∀x ∈ H , where A is a nonnegative operator and B is a complex symmetric operator on a separable complex Hilbert space H. Along the way, we also prove a new factorization theorem for complex symmetric operators. Mathematics Subject Classification (2000). 47B99.

In this paper, we prove the existence and uniqueness of squaremean asymptotically almost automorphic mild solution of a class of partial nonlocal neutral stochastic functional integro-differential equations with resolvent operators in a real separable Hilbert space. An example illustrating our main result is given.

We give a characterization of operators on a separable Hilbert space of norm less than one that can be represented as products of orthogonal projections and give an estimate on the number of factors. We also describe the norm closure of the set of all products of orthogonal projections.