ON the operatore quation ax + xb = q
Consider the operator equation (*) 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 (*). Sufficient conditions are given in order that (*) has a solution in the Schatten-von Neumann class Cp if Q is in Cp. Finally a sufficient condition for solvability of (*) is given in terms of T. Kato's notion of smoothness. © 1978 American Mathematical Society.
Proceedings of the American Mathematical Society
Goldstein, J. (1978). ON the operatore quation ax + xb = q. Proceedings of the American Mathematical Society, 70 (1), 31-34. https://doi.org/10.1090/S0002-9939-1978-0477836-2