Estimates for Solutions of the Operator Equation $\sum\limits_{{j = 1}}^{m} {{{A}_{j}}Q{{B}_{j}} = U}$

Abstract

Let X and Y be complex Banach spaces and let L(Y,X) denote the Banach space of all continuous linear operators from Y into X equipped with the uniform operator topology. If X = Y, then L(Y,X) is denoted simply by L(X). The aim of this note is to suggest an approach which provides estimates for ‖ Q ‖ where Q is a solution of the equation

$$\sum\limits_{{j = 1}}^{m} {{{A}_{j}}Q{{B}_{j}} = U} .$$

(1)

Here m is a positive integer, ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}} = [{{A}_{1}}, \ldots {{A}_{m}}]({\text{resp}}{\text{.}} {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{B}} = [{{B}_{1}}, \ldots ,{{B}_{m}}])$ is a commuting m-tuple of elements form L(X) (resp. L(Y)) and U ε L(Y,X).