Let X, Y be Banach spaces, S: X → Y and R: Y → X be bounded operators. We investigate common spectral properties of RS and SR. We then apply the result obtained to extensions, Aluthge transforms and upper triangular operator matrices.
Let R and S be two operators on a Hilbert space. We discuss the link between the subscalarity of RS and SR. As an application, we show that backward Aluthge iterates of hyponormal operators and p-quasihyponormal operators are subscalar.
Let be a Banach space and be a bounded linear operator on . We denote by the set of all complex such that does not have the single-valued extension property at . In this note we prove equality up to between the left Drazin spectrum, the upper semi-B-Fredholm spectrum and the semi-essential approximate point spectrum. As applications, we investigate generalized Weyl’s theorem for operator matrices and multiplier operators.