Some results on the quasisimilarity of operators (Q1179325)

Two bounded linear operators \(A\) and \(B\) on a complex Hilbert space \(H\) are said to be quasisimilar if there exist injective continuous linear operators \(X\) and \(Y\) on \(H\), each having dense range, such that \(AX=XB\) and \(YA=BY\). The aim of this note is to present a number of results concerning intersection properties of the spectra of quasisimilar operators. Two sample results are as follows. Given an operator \(T\), let \(\sigma_ B(T)\) denote the subset of the spectrum of \(T\) obtained by removing all isolated eigenvalues of finite algebraic multiplicity, let \(\sigma_ c(T)\) denote the continuous spectrum of \(T\), and let \(\sigma_ \ell(T)\) and \(\sigma_ r(T)\) denote respectively the left and right spectrum of \(T\). Suppose that \(A\) and \(B\) are quasisimilar. Then (i) if \(\tau\) is a connected component of \(\sigma_ \ell(B)\), then \(\tau\cap\sigma_ r(A)\) is non-empty; (ii) if \(V_ B\) is a non-empty closed subset of \(\sigma_ B(B)\) containing \(\sigma_ c(B)\), then \(\tau\cap\sigma_ B(A)\) is non-empty for every connected component \(\tau\) of \(V_ B\).