On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

摘要

A class U_k1 (J){mathcal{U}}_{kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q mvf¢s S_kp ×qp times q , {rm mvf's}, {mathcal{S}}_{kappa}^{p times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation T_W based on W and applied to S_k2p ×q{mathcal{S}}_{kappa 2}^{p times q}. Factorization formulas for mvf’s W in a subclass U°_k1 (J) of U_k1(J){mathcal{U}^{circ}_{kappa 1}} (J), {rm of}, {mathcal{U}}_{kappa 1}(J) found and then used to parametrize the set S_k1+k2p ×q ÇT_W [ S_k2p ×q ]{mathcal{S}}_{{kappa 1}+{kappa 2}}^{p times q} cap T_{W} left[ {mathcal{S}}_{kappa 2}^{p times q} right]. Applications to bitangential interpolation problems in the class S_k1+k2p ×q{mathcal{S}}_{{kappa 1}+{kappa 2}}^{p times q} will be presented elsewhere.