Zhong, Yong.2014-10-2119921992AAINN80204http://hdl.handle.net/10222/55340The 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.Mathematics.text