 ## Doctoral Dissertations

Fall 1994

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

#### Abstract

Consider the sets ${\cal P}\sb{\cal H}$ and ${\cal P}\sb{\cal K}$ of the projections onto closed subspaces of Hilbert spaces ${\cal H}$ and $\cal K$ respectively. From the usual partial orders (based upon set containment) on $\cal P\sb{\cal K}$ and $\cal P\sb{\cal H}$, we can define a partial order on $\cal P\sb{\cal K}\times\cal P\sb{\cal H}$ by ($Q\sb1,P\sb1)\le(Q\sb2, P\sb2)$ if and only if $P\sb1\le P\sb2$ and $Q\sb2\le Q\sb1.$ Then the map $\alpha : \cal P\sb{\cal K}\times \cal P\sb{\cal H}\to \cal P\sb{\cal K\oplus\cal H}$ given by $\alpha(Q,P)=(1-Q)\oplus P$ is an order-preserving map. In particular, if $\cal L\subseteq\cal P\sb{\cal K\times\cal H}$ is a lattice, then the restriction of $\alpha$ to $\cal L$ is a lattice isomorphism.

Let Alg(${\cal L}$) be the set of all operators taking each element of ${\cal L}$ into itself. If an operator T = $\left\lbrack\matrix{A&B\cr C&D\cr}\right\rbrack$ in ${\cal B(H,K)}$ belongs to Alg($\alpha({\cal L}))$, then for every pair ($Q,P$) in the lattice ${\cal L}$, (1) $QA(l-Q)= 0;$ (2) $QBP = 0;$ (3) $(1-P)C(1-Q)= 0;$ and (4)$(1-P)DP = 0.$.

The upper left coordinate of T must belong to $Alg(\{Q : (Q, P)\in {\cal L}\})$ and the lower right coordinate must belong to $Alg(\{P : (Q, P)\in {\cal L}\}).$ The upper right coordinate belongs to the subspace denoted below:$${\cal S(L)}= \{T\in B({\cal H,K}): QTP =0\ \forall\ (Q,P)\in {\cal L}\}.$$.

In this paper, we will investigate the subspace ${\cal S(L)}$ for various types of lattices ${\cal L}$. Using the above map $\alpha,$ we consider Alg($\alpha({\cal L}))$ as a subspace of two by two operator-valued matrices with ${\cal S(L)}$ imbedded in the upper right corner.

We will consider what properties transfer between a subspace of operator-valued two by two matrices and the subspaces derived from each component; we may use these results to compare Alg($\alpha({\cal L}))$ and ${\cal S(L)}.$.

In the following sections, we examine particular types of lattices ${\cal L}$, namely nests and commutative subspace lattices. Many properties of the associated algebras can be used to describe the subspace ${\cal S}(L).$.

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