Explicit solution of the operator equation A*X+X*A=BA*X+X*A=B

In this paper we find the explicit solution of the equationA*X+X*A=BA*X+X*A=Bfor linear bounded operators on Hilbert spaces, where X is the unknown operator. This solution is expressed in terms of the Moore-Penrose inverse of the operator A. Thus, results of J. H. Hodges [Some matrix equations over a finite field, Ann. Mat. Pura Appl. 44 (1957) 245–550] are extended to the infinite dimensional settings.