Interpolation: Operator-valued Case

Abstract

In classical Schur analysis, operator-valued functions appear naturally in a number of different settings, of which we mention:

• 1. The characteristic operator function, when the imaginary part of the operator is not of finite rank (but possibly trace class, or of Hilbert–Schmidt class). See for instance [110, 111, 242].

• 2. The time-varying case, when the complex numbers are replaced by diagonal operators. See [40, 161, 162, 163].

• 3. Interpolation in the Hardy space of the polydisk. One can reduce the problem to an operator-valued problem in one variable. The values are then assumed to be Hilbert-Schmidt operators. See [23, 24].