| dc.description | 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. | en_US |
