Invertible solutions to the operator equation $TA-BT=C$

If X is finite-dimensional, it is well known that for any C, a unique solution T of (1) exists provided that the eigenvalues of A are distinct from the eigenvalues of B [1]. An extension of this result has been given by Rosenblum [2 ]. For an arbitrary Banach space the operator equation (1) possesses a unique solution T provided that the spectrum of A is disjoint from the spectrum of B. Certain results concerning the invertibility of T are available in the special case where X is finite-dimensional and (1) is replaced by