Approximate unitary equivalence to skew symmetric operators
A skew symmetric operator is a bounded linear operator \(T\) on a complex separable infinite dimensional Hilbert space \(\mathcal{H}\) which satisfies the relation \(CTC=-T^*\) for some conjugation \(C\) on \(\mathcal{H}\). In [\textit{C. G. Li} and the author, Proc. Am. Math. Soc. 141, No. 8, 2755--2762 (2013; Zbl 1279.47040)], the authors observed that, for a skew symmetric normal operator \(T\), \(T|_{(\ker T)^\perp}\) can be represented (up to unitary equivalence) as \(N\oplus(-N)\), where \(N\) is a normal operator. It is the aim of the paper under review to classify another class of skew symmetric operators. More precisely, it is shown that a skew symmetric operator \(T\) satisfying \(C^*(T)\cap\mathcal{K}(\mathcal{H})=\{0\}\) is approximately unitarily equivalent to an operator of the form \(A\oplus(-A^t)\) for a certain operator \(A\) on \(\mathcal{H}\) (here \(C^*(T)\) denotes the \(C^*\)-algebra generated by \(T\) and the identity operator on \(\mathcal{H}\), \(\mathcal{K}(\mathcal{H})\) is the ideal of compact operators on \(\mathcal{H}\), while \(A^t\) is a transpose of \(A\), i.e., \(A^t=CA^*C\) for some conjugation \(C\) on \(\mathcal{H}\)). As an application, the author obtains a similar classification for unilateral weighted shifts with nonzero weights.
scientific article; zbMATH DE number 6381583
scientific article