On scalar extensions and spectral decompositions of complex symmetric operators (Q719553)

The landmark contribution of \textit{E. Bishop} [``A duality theorem for an arbitrary operator'', Pac. J. Math. 9, 379--397 (1959; Zbl 0086.31702)] opened a full new line of discoveries linking spectral decomposition properties of a linear (bounded) operator \(T\) and semi-local properties of the analytic function \(zI-T\). In particular, Bishop's conditions \((\beta)\) and \((\delta)\) were related to the existence of decomposable extensions, respectively co-extensions. The note under review exploits such well-known facts from local spectral theory, in conjunction with the class of complex symmetric operators. Although the results are not surprising, the note brings a lucid and necessary contribution to the modern theory of linear operators.