Asymptotic reflexivity
Let \(\mathcal{S}\) be a subspace of the space \(B(H)\) of all bounded linear operators on a Hilbert space~\(H\). A new reflexive hull \(\text{AsymRef}(\mathcal{S})\) of \(\mathcal{S}\) is introduced in the following manner: an operator \(T\in B(H)\) belongs to \(\text{AsymRef}(\mathcal{S})\) precisely when \(\lim(Te_{\lambda},f_{\lambda})\rightarrow 0\) whenever \(\{e_{\lambda}\}\) and \(\{f_{\lambda}\}\) are nets of vectors of \(H\) such that \((Se_{\lambda},f_{\lambda})\rightarrow 0\) for every \(S\in\mathcal{S}\). The subspace \(\mathcal{S}\) is called asymptotically reflexive if \(\mathcal{S}=\text{AsymRef}(\mathcal{S})\). An operator \(T\in B(H)\) is called asymptotically reflexive if \(\text{AsymRef}(\mathbb{C}T)\) equals the algebra \(\mathcal{P}(T)\) of all polynomials on \(T\). The notion of asymptotic reflexivity is related to, but distinct from, the notion of approximate reflexivity where boundedness of the nets \(\{e_{\lambda}\}\) and \(\{f_{\lambda}\}\) is required. In particular, asymptotically reflexive spaces need not be norm closed. The author establishes a number of properties of the asymptotic reflexive hull showing, in particular, that \(\text{AsymRef}(\bigcup_{\alpha}\mathcal{S_{\alpha}}) = \bigcup_{\alpha}\text{AsymRef}(\mathcal{S_{\alpha}})\) for any upward directed family \(\{\mathcal{S_{\alpha}}\}\) of subspaces of \(B(H)\). He shows that every operator is asymptotically reflexive and completely characterises the asymptotic reflexive hull of \(\mathcal{S}\) in terms of the subspace of finite rank operators in \(\mathcal{S}\). In particular, he establishes the formula \(\text{AsymRef}(\mathcal{S}) = \mathcal{S} + \text{AsymRef}(\mathcal{S}\cap\mathcal{F})\), where \(\mathcal{F}\) is the set of all operators on \(H\) of finite rank. A beautiful analogue of the characterisation of hereditarily reflexive subspaces due to \textit{A.\,I.\thinspace Lo\-gi\-nov} and \textit{V.\,S.\thinspace Sul'man} [Math.\ USSR, Izv.\ 9 (1975), 1189--1201 (1977; Zbl 0339.46047)] is also established: an asymptotically reflexive subspace \(\mathcal{S}\) has the property that every subspace of \(\mathcal{S}\) is asymptotically reflexive if and only if every linear functional on \(\mathcal{S}\) is the pointwise limit of vector functionals. The paper finishes with a list of open questions.
scientific article; zbMATH DE number 5141638
scientific article