Quaternionic de Branges Spaces and Characteristic Operator Function by Daniel Alpay & Fabrizio Colombo & Irene Sabadini

Author:Daniel Alpay & Fabrizio Colombo & Irene Sabadini
Language: eng
Format: epub
ISBN: 9783030383121
Publisher: Springer International Publishing

Thus

i.e., the resolvent operator is unitarily equivalent to the backward-shift operator . We note also that by Stone’s theorem when . This argument cannot be extended in a direct way when are not real.

Let be a subspace of . If , and we have

This can be written as

We consider a (possibly) unbounded closed operator T for which there exists such that the elements

are dense in where is an open set in not intersecting the S-spectrum of T. When T is bounded, this condition is equivalent to ask that u is cyclic for , that is the span of the vectors is dense in .

We denote by G the linear operator and consider the function

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