Decomposability and structure of bands of nonnegative operators.
Date
1996
Authors
Marwama, Alka.
Journal Title
Journal ISSN
Volume Title
Publisher
Dalhousie University
Abstract
Description
The Invariant Subspace Problem is one of the most intriguing problems in Hilbert Space Theory. Attempts to solve it have led to other interesting related problems in Operator Theory. In the past few years extensive research has been done to find conditions under which a semigroup of operators (i.e., a collection of operators closed under multiplication) can be shown to have a common nontrivial invariant subspace. Such a semigroup is called reducible.
The present thesis focuses on semigroups of (functionally) nonnegative operators and in particular, semigroups of nonnegative idempotents called nonnegative bands on a finite or infinite-dimensional Hilbert space and obtains necessary and sufficient conditions for the existence of special kind of invariant subspaces for these semigroups which are termed standard subspaces. (an n x n matrix with nonnegative entries is an example of a nonnegative operator on $\doubc\sp{n}$ and the span of a subset of the standard basis $\{e\sb1,e\sb2,\...,e\sb{n}\}$ of $\doubc\sp{n}$ is a standard subspace of $\doubc\sp{n}$). A semigroup with a common nontrivial standard invariant subspace is said to be decomposable. It is proved that a nonnegative band with each member having rank greater than one and containing at least one finite-rank operator is decomposable. An example of an indecomposable nonnegative band in ${\cal B}(l\sp2)$ with constant infinite rank is given and it is shown that infiniteness of such a band makes it decomposable. Further, the structure of constant finite-rank bands is studied. Under a special condition of fullness, maximal nonnegative bands of constant rank r are shown to be the direct sum of r maximal rank-one indecomposable nonnegative bands. Finally, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained, which in view of the result stated above, gives a geometric characterization of maximal, finite-rank, indecomposable, nonnegative bands.
Thesis (Ph.D.)--Dalhousie University (Canada), 1996.
The present thesis focuses on semigroups of (functionally) nonnegative operators and in particular, semigroups of nonnegative idempotents called nonnegative bands on a finite or infinite-dimensional Hilbert space and obtains necessary and sufficient conditions for the existence of special kind of invariant subspaces for these semigroups which are termed standard subspaces. (an n x n matrix with nonnegative entries is an example of a nonnegative operator on $\doubc\sp{n}$ and the span of a subset of the standard basis $\{e\sb1,e\sb2,\...,e\sb{n}\}$ of $\doubc\sp{n}$ is a standard subspace of $\doubc\sp{n}$). A semigroup with a common nontrivial standard invariant subspace is said to be decomposable. It is proved that a nonnegative band with each member having rank greater than one and containing at least one finite-rank operator is decomposable. An example of an indecomposable nonnegative band in ${\cal B}(l\sp2)$ with constant infinite rank is given and it is shown that infiniteness of such a band makes it decomposable. Further, the structure of constant finite-rank bands is studied. Under a special condition of fullness, maximal nonnegative bands of constant rank r are shown to be the direct sum of r maximal rank-one indecomposable nonnegative bands. Finally, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained, which in view of the result stated above, gives a geometric characterization of maximal, finite-rank, indecomposable, nonnegative bands.
Thesis (Ph.D.)--Dalhousie University (Canada), 1996.
Keywords
Mathematics.