On the closure of the complex symmetric operators: compact operators and weighted shifts (Q1937460)

Let \({\mathcal B}({\mathcal H})\) denote the algebra of all bounded linear operators on a separable, infinite-dimensional, complex Hilbert space \({\mathcal H}\). An operator \(T \in {\mathcal B}({\mathcal H})\) is said to be complex symmetric if there exists a conjugation \(C\) (that is, a conjugate-linear, isometric involution on \({\mathcal H}\)) such that \(T=CT^\ast C\). The family of all complex symmetric operators is denoted by CSO. The paper under review investigates the structure of the closure of CSO. In Section~2 it is shown that the closure of CSO in the strong-\(\ast\) topology is the whole \({\mathcal B}({\mathcal H})\). As a consequence, the same thing is true for the strong operator topology and the weak operator topology. Hence, the closure of CSO is only of interest if the closure is taken with respect to the norm topology on \({\mathcal B}({\mathcal H})\). In Section~3, it is shown that, if \(T \in \overline{\text{CSO}}\) and \(T\) is compact, then \(T \in CSO\). A consequence of this result is a simple proof that \(\overline{\text{CSO}} \neq {\mathcal B}({\mathcal H})\). A typical example of an operator in \({\mathcal B}({\mathcal H}) \backslash \overline{\text{CSO}}\) is the Kakutani shift. The weight sequence \(\{\alpha_n\}_{n=1}^\infty\) of the Kakutani shift has a subsequence that converges to zero and another one that tends to \(\sup \{\alpha_n\}_{n=1}^\infty\). In Section~4, it is shown that this behavior is shared by all irreducible weighted shifts in \( \overline{\text{CSO}}\). In Section~5, it is shown that every irreducible weighted shift that is approximately Kakutani belongs to the family \(\overline{\text{CSO}} \backslash \text{CSO}\). This allows the authors to construct operators in \(\overline{\text{CSO}} \backslash \text{CSO}\) whose moduli have desired spectral properties.