Let T be a self-adjoint operator acting in a separable Hilbert space H. We establish a correspondence between the reducing subspaces of T that come from a spectral projection and the convex, norm-closed bands in the set of finite Borel measures on R. IfH is not separable, we still obtain a reducing subspace corresponding to each convex norm-closed band. These observations lead to a unified trea...

The aim of the paper is to provide some regularity results for stochastic convolutions corresponding to stochastic Volterra equations in separable Hilbert space. We study convolutions of the form WΨ(t) := ∫ t 0 S(t − τ)Ψ(τ)dW (τ), t ≥ 0, where S(t), t ≥ 0, is so-called resolvent for Volterra equation considered,Ψ is an appropriate process and W is a cylindrical Wiener process. In the paper we e...

In this article we define some difference sequence spaces using a new difference operator. We show that these spaces can be made BK-spaces under a suitable norm. We also find their isometrically isomorphic spaces and thus we find the dual of some of the spaces. Furthermore we study the spaces for separable space, reflexive space, Hilbert space and investigate for solid space, symmetric space, m...

We show that the space of all Lelek fans in a Cantor fan, equipped with the Hausdorff metric, is homeomorphic to the separable Hilbert space. This result is a special case of a general theorem we prove about spaces of upper semicontinuous functions on compact metric spaces that are strongly discontinuous.

A Wiener-Hopf operator on a Banach space of functions on R is a bounded operator T such that PS−aTSa = T , a ≥ 0, where Sa is the operator of translation by a. We obtain a representation theorem for the Wiener-Hopf operators on a large class of functions on R with values in a separable Hilbert space.

With every E0–semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0–semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and a...

A sequence ${T_n}_{n=1}^{infty}$ of bounded linear  operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the disk-scaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C}, | alpha | leq 1}cap M$ is dense in $M$. The goal of t...

Kadison and Kastler introduced a natural metric on the collection of all C∗-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, obtaining embeddings from certain near inclusions inv...

In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space $(\mathcal{P}_2(\mathsf{H}),W_2)$ Borel probability measures with finite quadratic moment on a separable Hilbert $\mathsf{H}$.