Lossless Inverse Scattering and Reproducing Kernels for Upper Triangular Operators

In this paper the topics mentioned in the title are studied in the algebra of upper triangular bounded linear operators acting on the space l N 2 of “square summable” sequences f = (..., f −1, f 0, f l,...) with components f i in a complex separable Hilbert space N. Enroute analogues of point evaluation (where in this case the points are operators and the functions are operator valued functions of operators) and simple Blaschke products are developed in this more general context. These tools are then used to establish a theory of structured reproducing kernel Hilbert spaces in the class of upper triangular Hilbert Schmidt operators on l N 2. (An application of these tools and spaces to solve a Nevanlinna Pick interpolation problem wherein both the interpolation points and the values assigned at those points are block diagonal operators, will be considered in a future publication.)

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