Operator matrices: SVEP and Weyl's theorem (Q2504864)

A bounded linear operator \(A\) on a Hilbert space \(H\) is said to be a Weyl operator if it is a Fredholm operator with index equal to 0. The Weyl spectrum of \(A\), denoted by \(\sigma_w(A)\), is the set of all complex numbers \(\lambda\) such that \((\lambda I -A)\) is not a Weyl operator, and the operator \(A\) is said to satisfy property \((W)\) if the spectrum of \(A\) consists of the Weyl spectrum together with the isolated eigenvalues of \(A\) having finite-dimensional eigenspaces. An operator \(A\) has the single-valued extension property (SVEP) at the complex number \(\lambda_0\) if, for every open disc \(D\) centered at \(\lambda_0\), the only analytic function \(f:D\to H\) that satisfies \((\lambda I-A)f(\lambda)=0\) for all \(\lambda\) in \(D\) is the function \(f\equiv 0\). For instance, \(A\) has SVEP at every isolated point of its spectrum. An operator is simply said to have SVEP if it has SVEP at every complex number. The present paper, which builds on earlier work of the authors and of W.\,Y.\thinspace Lee, among others, explores conditions on operators \(A\) and \(B\) that ensure that \(A\oplus B\) has property \((W)\) or, more generally, that the operator matrix \(M_C=\begin{pmatrix} A&C\\ 0&B \end{pmatrix}\) has property \((W)\) for every \(C\). Among their results on these matters, Theorem~2.5 demonstrates that, if \(A\) and \(B\) have the SVEP, \(A\) is isolate, and both \(A\) and \(A\oplus B\) satisfy property \((W)\), then \(M_C\) also has property \((W)\) for all \(C\). Theorem~2.6 shows that, if \(A\) and \(B\) are isoloid operators with the SVEP and both satisfy property \((W)\), then \(M_C\) has property \((W)\) for all \(C\). Corollary~2.12 provides a host of examples. Namely, if \(T=A\oplus B\) is the decomposition into normal and pure parts of an operator \(T\) from any of the classes \(p\)-hyponormal, quasi-\(p\)-hyponormal, \(M\)-hyponormal, \(k\)-quasihyponormal, or paranormal, then the corresponding \(M_C\) satisfies property \((W)\) for all \(C\). Also, Theorem~2.14 gives necessary and sufficient conditions for an operator \(A\) to satisfy \((W)\) in the case that either \(A\) or \(A^*\) has the SVEP. The present paper makes a worthwhile contribution to the substantial body of work on Weyl's theorem and related issues.