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Operator Weighted Shifts on Hardy Spaces. Edition No. 1
VDM Publishing House, Sep 2010, Pages: 88
There are several classes of operators defined and studied on the Hardy space of which shift operators are an important class. In 1949, Beurling characterized the invariant subspaces of the unilateral shift operator. A natural generalization of scalar weighted shift is the operator weighted shift which is our object of study. We define operator weighted shifts on Hardy Spaces over a separable complex Hilbert space. Following this the spectrum, the invariant subspaces and the (minimal) reducing subspaces of these operators are determined and analyzed. A discussion on hypercyclic operator weighted shifts and their characterization in terms of their weight sequences is included. Finally 'The Subnormal completion problem' in the context of operator weighted shifts of finite multiplicity is addressed. A major obstacle to progress in Operator Theory is the dearth of concrete examples whose properties can be explicitly determined. The present work brings out many significant properties of operator weighted shifts which will help generate a huge repertoire of examples that can be used to validate important theoretical results of Operator Theory.
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