Functional positivity and invariant subspaces of semigroups of operators.
Date
1992
Authors
Zhong, Yong.
Journal Title
Journal ISSN
Volume Title
Publisher
Dalhousie University
Abstract
Description
The main results in this thesis are about multiplicative semigroups of functionally positive operators and their invariant subspaces.
Let X be a topological space, and with its Borel structure, a standard Borel space, and m a $\sigma$-finite regular Borel measure on X such that ${\cal L}\sp2(X,m$) is called (functionally) positive if it maps non-negative functions to non-negative functions. Generally, the algebra generated by all positive operators is not closed in operator norm topology. We introduce a new norm on the algebra and show, using classical methods of functional analysis, that the algebra is a Branach $\*$-algebra under the new norm. The spectral aspects of elements of the Branach algebra are discussed.
Suppose ${\cal S}$ is a semigroup of positive integral operators on ${\cal L}\sp2(X,m$). We show by analyzing the structure of the kernels that ${\cal S}$ has a non-trivial invariant subspace if every operator in ${\cal S}$ is quasinilpotent. We construct a special kind of bases of the ranges of positive integral idempotent operators consisting of only non-negative functions. Using these bases, we prove that ${\cal S}$ has a non-trivial invariant subspace if it contains a non-zero compact operator and $r(AB)\le r(A)r(B)$ for all $A,B$ in ${\cal S}$. Also, we prove that if ${\cal S}$ is a semigroup of positive integral operators with the kernels satisfying certain positivity conditions, then there exists a non-trivial standard subspace, i.e., a subspace of the form $X\sb{E}{\cal L}\sp2(X,m)$ for some Borel set E in X, invariant under ${\cal S}$. We give a non-compact analogue of the Lomonosov - de Pagter result. Let T be an injective positive quasinilpotent operator dominating a non-zero compact positive operator $T\sb0$, i.e., $T$ $-$ $T\sb0$ is positive. Assume ${\cal C}$ is a collection of positive operators contained in a norm-closed algebra ${\cal A}$ with ${\cal A}T\subseteq T{\cal A}$. Then there exists a non-trivial standard subspace invariant under ${\cal C}$ and T.
Finally, we construct a semigroup of positive nilpotent operators with no non-trivial invariant subspaces.
Thesis (Ph.D.)--Dalhousie University (Canada), 1992.
Let X be a topological space, and with its Borel structure, a standard Borel space, and m a $\sigma$-finite regular Borel measure on X such that ${\cal L}\sp2(X,m$) is called (functionally) positive if it maps non-negative functions to non-negative functions. Generally, the algebra generated by all positive operators is not closed in operator norm topology. We introduce a new norm on the algebra and show, using classical methods of functional analysis, that the algebra is a Branach $\*$-algebra under the new norm. The spectral aspects of elements of the Branach algebra are discussed.
Suppose ${\cal S}$ is a semigroup of positive integral operators on ${\cal L}\sp2(X,m$). We show by analyzing the structure of the kernels that ${\cal S}$ has a non-trivial invariant subspace if every operator in ${\cal S}$ is quasinilpotent. We construct a special kind of bases of the ranges of positive integral idempotent operators consisting of only non-negative functions. Using these bases, we prove that ${\cal S}$ has a non-trivial invariant subspace if it contains a non-zero compact operator and $r(AB)\le r(A)r(B)$ for all $A,B$ in ${\cal S}$. Also, we prove that if ${\cal S}$ is a semigroup of positive integral operators with the kernels satisfying certain positivity conditions, then there exists a non-trivial standard subspace, i.e., a subspace of the form $X\sb{E}{\cal L}\sp2(X,m)$ for some Borel set E in X, invariant under ${\cal S}$. We give a non-compact analogue of the Lomonosov - de Pagter result. Let T be an injective positive quasinilpotent operator dominating a non-zero compact positive operator $T\sb0$, i.e., $T$ $-$ $T\sb0$ is positive. Assume ${\cal C}$ is a collection of positive operators contained in a norm-closed algebra ${\cal A}$ with ${\cal A}T\subseteq T{\cal A}$. Then there exists a non-trivial standard subspace invariant under ${\cal C}$ and T.
Finally, we construct a semigroup of positive nilpotent operators with no non-trivial invariant subspaces.
Thesis (Ph.D.)--Dalhousie University (Canada), 1992.
Keywords
Mathematics.