Operator-valued typically real functions induced by a contraction (Q1289044)

Let \(H\) be a complex Hilbert space and let \(L(H)\) denote the algebra of all bounded linear operators on \(H\). Let \(F(z)= \sum_{n=0}^\infty z^nA_n\) be an operator-valued analytic function whose coefficients are bounded operators in \(L(H)\) for \(z\) in the open unit disc \(D\) and the series is convergent in the strong operator topology. In this paper, we discuss operator-valued typically real functions \(F(z)= \sum_{n=0}^\infty z^nA_n\) which generalize complex-valued typically real functions. We characterize operator-valued typically real functions and study such functions \(F(z)\) induced by a contraction on \(H\). In addition, we consider \(m\times n\)-tuple operator-valued typically real functions and positively real functions. Finally, we characterize operator-valued typically real functions in the finite-dimensional cases.