Sylvester equations and polynomial separation of spectra
Olavi Nevanlinna · Operators and Matrices · 2019
Sylvester equations AX -XB = C have unique solutions for all C when the spectra of A and B are disjoint. Here A and B are bounded operators in Banach spaces. We discuss the existence of polynomials p such that the spectra of p(A) and p(B) are well separated, either inside and outside of a circle or separated into different half planes. Much of the discussion is based on the following inclusion sets for the spectrum: V p (T ) = { C : |p( )| p(T ) } where T is a bounded operator. We also give an explicit series expansion for the solution in terms of p(M) , where M = A C B , in the case where the spectra of A and B lie in different components of V p (M) .