On reflexivity of operators
For a bounded operator T on a Hilbert space H, let \(A_ T(W_ T)\) denote the weak-star closed (WOT closed) algebra generated by T and the identity. The operator T is said to be reflexive if whenever Lat \(T\subset Lat A\) then \(A\in W(T)\). Recall that \(A_ T\) is the dual of the quotient space \(Q_ T=C_ 1(H)/_{\perp_{A_ T}}\), where \(C_ 1(H)\) is the Banach space of trace class operators. \textit{H. Bercovici, C. Foias}, and \textit{C. Pearcy} [CBMS Reg. Conf. Ser. Math. 56, XI, 108 p. (1985; Zbl 0569.47007)] have defined the properties \((B_{m,n})\) and \((\tilde B_{m,n})\). We say that \(A_ T\) has property \((B_{m,n})\) if every \(m\times n\) system \[ (1)\quad \{[L_{ij}]\in Q_ T: 1\leq i\leq m,\quad 1\leq j\leq n\} \] can be written as \([L_{ij}]=[x_ i\otimes y_ j]\) with \(x_ i\) and \(y_ j\) belonging to H. We say that \(A_ T\) has property \((\tilde B_{m,n})\) if whenever (1) has an approximate solution \([x_ i'\otimes y_ j']\) it has an exact solution \([L_{ij}]=[x_ i\otimes y_ j].\) The above properties are used in determining whether an operator is reflexive. \textit{J. A. Deddens} and \textit{P. A. Fillmore} [Linear Algebra Appl. 10, 89-93 (1975; Zbl 0301.15011)], \textit{D. W. Hadwin} and \textit{E. A. Nordgren} [J. Oper. Theory 7, 3-23 (1982; Zbl 0483.47023)], have given necessary and sufficient conditions that an algebraic operator be reflexive. In the article under review it is proved that if T is an algebraic operator such that \(A_ T\) has property \((B_{2,2})\) then T is reflexive. They also give an example to show that having property \((B_{1,2})\) does not, in general, imply the possession of the property \((\tilde B_{1,2})\), thereby settling a question raised in an earlier work.
scientific article; zbMATH DE number 4032449
scientific article