A criterion for reducibility of certain representations of abelian groups is established. Among the applications of this criterion, we give a positive answer to the translation invariant subspace problem for weighted spaces on locally compact abelian groups, for even weights and .
Let S(X) denote the set of all closed subsets of a topological space X, and C(X) the set of all continuous mappings f:X → X. A family 𝓐 ⊆ S(X) is called reflexive if there exists ℱ ⊆ C(X) such that 𝓐 = {A ∈ S(X): f(A) ⊆ A for every f ∈ ℱ}. We investigate conditions ensuring that a family of closed subsets is reflexive.
In this note we deal with a version of James' Theorem for numerical radius, which was already considered in [4]. First of all, let us recall that this well known classical result states that a Banach space satisfying that all the (bounded and linear) functionals attain the norm, has to be reflexive [16].
We study reflexivity of bilattices. Some examples of reflexive and non-reflexive bilattices are given. With a given subspace lattice we may associate a bilattice . Similarly, having a bilattice we may construct a subspace lattice . Connections between reflexivity of subspace lattices and associated bilattices are investigated. It is also shown that the direct sum of any two bilattices is never reflexive.
We prove that any set of commuting isometries on a separable Hilbert space is reflexive.
The purpose of this paper is to provide a method of reduction of some problems concerning families of linear operators with domains to a problem in which all the operators have the same domain . To do it we propose to construct a family of automorphisms of a given Banach space X having two properties: (i) the mapping is sufficiently regular and (ii) for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition with a parameter;...
In this paper, we will use results developed by Ansari and Enflo in the theory of bounded linear operators with dense range. We define two maps, with regards to some parameters, that control surjectivity default of a given operator, and prove analycity for the first one and global continuity for the other one. Minimisation results are also obtained in relation to this study.