Lyapunov equation for infinite-dimensional discrete bilinear systems (Q1175514)

Let \(H\) be a separable complex Hilbert space, \(B[H]\) the Banach algebra of all bounded linear operators in \(H\), \(B^ +[H]\) the elements of \(B[H]\) nonnegatives and selfadjoint, \(G[H]\) the group of invertible operators from \(B[H]\), and \(G^ +[H]=B^ +[H]\cap G[H]\). For every \(Q\in B[H]\) let \({\mathfrak F}(Q)=FQF^*\). The following result is known: \(r_ \sigma(F)<1\) iff for every \(V\in G^ +[H]\) there exists a unique solution \(W\in G^ +[H]\) of the (Lyapunov) equation \(V=W-{\mathfrak F}(W)\). The main result on the paper extends this property to a case with more general transformation \({\mathfrak F}\), and generalizes previous results of the authors.