The purpose of the present notes is to examine following issues related Chernoff estimate: (1) For contractions on a Banach space we modify $\sqrt{n}$-estimate and apply it in proof product formula for $C_0$-semigroups \textit{strong} operator topology. (2) We use idea {probabilistic} approach, proving estimate strong topology, uplift \textit{operator-norm} \textit{quasi-sectorial} contraction ...

Let X and Y be Banach spaces and (Ω,Σ, μ) a finite measure space. In this note we introduce the space L[μ;L (X, Y )] consisting of all (equivalence classes of) functions Φ : Ω 7→ L (X, Y ) such that ω 7→ Φ(ω)x is strongly μ-measurable for all x ∈ X and ω 7→ Φ(ω)f(ω) belongs to L(μ; Y ) for all f ∈ L ′ (μ;X), 1/p + 1/p = 1. We show that functions in L[μ;L (X, Y )] define operator-valued measures...

We show that if the set of all bounded strongly continuous cosine families on a Banach space X is treated as a metric space under the metric of the uniform convergence associated with the operator norm on the space L (X) of all bounded linear operators on X, then the isolated points of this set are precisely the scalar cosine families. By definition, a scalar cosine family is a cosine family wh...

Abstract In this paper, we will investigate some topological properties of Rb set of all bounded generalized Rhally matrices given by b sequences in H. We will show that Rb is closed subspace of B (H) and we will define an operator from T to Rb and show that this operator is injective and continuous in strong-operator topology.

An algebra of bounded linear operators on a Hilbert space is called strongly compact whenever each of its bounded subsets is relatively compact in the strong operator topology. The concept is most commonly studied for two algebras associated with a single operator T : the algebra alg(T ) generated by the operator, and the operator’s commutant com(T ). This paper focuses on the strong compactnes...

I.1 Two short problems: 1. Suppose that (X,A, μ) is a measure space, and X = ∪j=1Xj with Xj ⊂ Xj+1 for all j. Let χj be the indicator (characteristic) function of Xj, and let Mj denote the multiplication operator by χj acting on Z = Lp(X, dμ), 1 ≤ p < ∞. Show that Mj → I, the identity operator, in the strong operator topology, i.e. the weakest topology on bounded linear operators L(Z) in which ...