#### Recommended Citation

*Dissertation*, January 1, 1993, pages 1-72.

*NOTE: At the time of publication, the author Mark Stankus was not yet affiliated with Cal Poly*.

#### Abstract

We explore the elementary operator theory of the equation

(0.1) ∑ *c _{m,n}T^{*n}T^{m}*=0

for

*c*E C,

_{m,n}*c*nonzero for only finitely many

_{m,n}*m, n*and

*T*a bounded linear transformation on a complex Hilbert space in Chapters 1 and 2. We explore the equation

(0.2) *T ^{*2}T - T^{*}T^{2} + T - T^{*}* = 0

in greater depth in Chapters 4 and 5.

Chapter 1 explores the algebraic and *C*^{*}-algebraic aspects of the equation (0.1) and both the spectral picture of and growth conditions on the resolvent of the operator *T* satisfying (0.1).

Chapter 2 explores the implications of Rosenblum's Theorem to the study of (0.1). These implications are sufficient in some cases to completely classify a solution to (0.1) given information about the spectrum of *T*. Chapter 2 also recalls a few definitions and results from the theory of von Neumann algebras which will be used in the rest of the paper.

Chapter 3 guarantees the existence of maximal invariant subspaces *M* for an operator *T* such that *T* restricted to M is a member of a fixed family of operators. This provides an approach to completely solving the equation (0.1) for *T* for certain choices of *c.*

In Chapters 4 and 5, we study operators *T* satisfying (0.2). These operators are termed *isosymmetries*. The results of Chapters 1, 2 and 3 *do not* solve equation (0.2).

Chapter 4 gives the elementary operator theory of isosymmetries.

Chapter 5 classifies several collections of isosymmetries. Indeed, if *T* is an isosymmetry and *T* is hyponormal, *T* is a contraction, *Im*(*T*) ≥ 0 or *Im*(*T*) ≤ 0, then *T* is subnormal and the minimal normal extension of *T* has the same properties. If *T ^{*}T* ≥ 1, then

*T*is the restriction to an invariant subspace of a direct integral of rank one perturbations of the unilateral shift. If the spectrum of

*T*equals its boundry, then

*T*has the form of a direct integral of 1 x 1 and 2 x 2 matrices. These constraints arise naturally from the analysis of Chapter 4.

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#### Disciplines

Mathematics

#### Copyright

1993 Mark Stankus

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**URL:** http://digitalcommons.calpoly.edu/math_fac/19