#### Date of Award

Spring 1986

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### Abstract

It is shown that Lat(, 1/2)(H('(INFIN))(S)), the invariant operator ranges of the commutant of the unilateral shift, is a proper sub-lattice of the lattice of invariant operator ranges of the unilateral shift, S. The notion of a strange operator range for S of order n where n (ELEM) is introduced and it is demonstrated that there exist strange ranges for S of every order. This is done by deriving an operator range condition which is sufficient to insure that a pair of quasi-similar compressions of shifts really be similar. A set of operator ranges which forms a sub-lattice of Lat(, 1/2)(H('(INFIN))(S)) is introduced, which is conjectured to be Lat(, 1/2)(H('(INFIN))(S)). The conjecture is shown to be equivalent to the assertion that the image of S under certain homomorphisms of H('(INFIN))(S) into B(H) is similar to a contraction.

It is proven that the ranges of operators from a commutative C*-algebra form a lattice under intersection and vector sum. If P and Q are projections in B(H) with non zero intersection and so that the angle between their ranges is 0, then it is shown that the ranges of the operators in the C*-algebra generated by P and Q does not contain the intersection of the ranges of P and Q. Thus, non-commutative C*-algebras need not have ranges which form a lattice. The question of whether the ranges of operators from different kinds of algebras form lattices is taken up and examples are provided.

It is proven that any pair of subspaces of a Hilbert space can be the ranges of a pair of commuting operators. A family of one dimen- sional subspaces of a Hilbert space, H, is shown to representable as the set of ranges of a family of commuting operators if and only if for each subspace the linear span of the union of the remaining sub-spaces is not dense in H. The sets of three subspaces of C('3) which can be the ranges of commuting operators are characterized.

#### Recommended Citation

ROY, CHARLES LUCIEN, "OPERATOR RANGES OF SHIFTS AND C*-ALGEBRAS (STRANGE RANGE, QUASI-SIMILARITY, LATTICE)" (1986). *Doctoral Dissertations*. 1484.

https://scholars.unh.edu/dissertation/1484