Operator-Valued Typically Real Functions Induced by a Contraction

Let H be a complex Hilbert space and let LH‘ denote the algebra of all bounded linear operators on H. Let Fz‘ D P1 nD0 znAn be an operator-valued analytic function whose coefficients are bounded operators in LH‘ for z in the open unit disc and the series is convergent in the strong operator topology. In this paper, we discuss operator-valued typically real functions Fz‘ D P1 nD0 znAn 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 n-tuple operator-valued typically real functions and positively real functions. Finally, we charaterize operator-valued typically real functions in the finite-dimensional cases