On the operator equation 𝐴𝑋+𝑋𝐵=𝑄

Abstract

Consider the operator equation ( ∗ ) A X + X B = Q (\ast )AX + XB = Q ; here A and B are (possibly unbounded) selfadjoint operators and Q is a bounded operator on a Hilbert space. The theory of one parameter semigroups of operators is applied to give a quick derivation of M. Rosenblum’s formula for approximate solutions of ( ∗ ) (\ast ) . Sufficient conditions are given in order that ( ∗ ) (\ast ) has a solution in the Schatten-von Neumann class C p {\mathcal {C}_p} if Q is in C p {\mathcal {C}_p} . Finally a sufficient condition for solvability of ( ∗ ) (\ast ) is given in terms of T. Kato’s notion of smoothness.