Intersection relations between the spectra of quasisimilar operators (Q1309132)

Let \(X\) and \(Y\) be infinite-dimensional complex Banach spaces, \(A: X\to X\), \(B: Y\to Y\) be continuous linear operators. \(A\) and \(B\) are said to be quasisimilar \((A\overset {q}\sim B)\) if there exist continuous linear injective operators \(P: X\to Y\) and \(Q: Y\to X\) with dense ranges, such that \(PA= BP\) and \(QB= AQ\). The author continues the study of the question of when certain parts of spectra of quasisimilar operators have non-empty intersections. The question has its origin in a result of \textit{T. B. Hoover} [Ill. J. Math. 16, 678-686 (1972; Zbl 0249.47014)], which says that if \(A\overset {q}\sim B\), then \(G(A)\cap G(B)\neq\emptyset\).