Browder essential approximate point spectra and hypercyclicity for operator matrices
Let \(H\) and \(K\) be infinite-dimensional separable Hilbert spaces and let \(B(H,K)\) denote the set of bounded linear operators from \(H\) to \(K\) (\(B(H)\) if \(K=H\)). If \(A\in B(H)\) and \(B\in B(K)\) are given, let \(M_C\) be the operator acting on \(H\oplus K\) of the form \(M_C= \left(\begin{smallmatrix} A & C\\0 & B\end{smallmatrix}\right)\) for some \(C\in B(K,H)\). In the paper under review, the author studies necessary and sufficient conditions on \(A\) and \(B\) for the existence of \(C\) with \(M_C\) upper semi-Fredholm, that is, with closed range \(R(M_C)\) and finite-dimensional kernel \(N(M_C)\). In particular, it is shown that if \(A\in B(H)\) is upper semi-Fredholm with \(\text{asc}(A):=\inf\{n\geq0:\) \(N(A^n)=N(A^{n+1})\}<\infty\) and \(d(A):=\dim(H/R(A))<\infty\) and \(B\in B(K)\), then there exists \(C\in B(K,H)\) such that \(M_C\) is upper semi-Fredholm with \(\text{asc}(M_C)<\infty\). In addition, the author also explores some hypercyclic properties of these operator matrices. In particular, under some assumptions on different kinds of spectra of \(A\) and \(B\), it is shown that \(M_0\) can be approximated (in the strong operator topology) by hypercyclic operators on \(H\oplus K\) if and only if the same holds for \(M_C\) for every \(C\in B(K,H)\). Related results can be found in two papers of the author, \textit{M.\,Z.\thinspace Guo} and \textit{B.\,Meng} [Acta Math.\ Sin., Engl.\ Ser.\ 22, No.\,1, 169--178 (2006; Zbl 1129.47014)] and the author and \textit{B.\,Meng} [J.~Math.\ Anal.\ Appl.\ 304, No.\,2, 759--771 (2005; Zbl 1083.47006)].
Browder essential approximate point spectrum
scientific article; zbMATH DE number 5190943