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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