On \(m\)-complex symmetric operators
An antilinear operator \(C\) acting on a complex separable Hilbert space \(H\) is said to be a conjugation if \(C^2=I\) and \(\langle Cx,Cy\rangle=\langle y,x \rangle\) for all \(x,y\in H\). A bounded linear operator \(T\) on \(H\) is a \(C\)-complex symmetric operator if \(T^*=CTC\). Motivated by the Helton class operators, the authors introduce the class of \(m\)-complex symmetric operator as a generalization of the class of complex symmetric operators: \(T\) is said to be \(m\)-complex symmetric if \[ \sum_{k=0}^m(-1)^{m-k}\begin{pmatrix} m\\k\end{pmatrix} T^{*k}CT^{m-k}C=0 \] for some positive integer \(m\). Then they study the spectral properties of \(m\)-complex symmetric operators. Among other things, they prove that, if \(T\) is \(m\)-complex symmetric with conjugation \(C\), then \(T^n\) is \(m\)-complex symmetric with the same conjugation for any \(n\in\mathbb{N}\). Some local spectral properties for \(m\)-complex symmetric operators are also investigated.
decomposable operators, single valued extension property
scientific article; zbMATH DE number 6620740