We explore the elementary operator theory of the equation
(0.1) ∑ cm,nT*nTm=0
for cm,n E C, cm,n nonzero for only finitely many m, n and Ta bounded linear transformation on a complex Hilbert space in Chapters 1 and 2. We explore the equation

(0.2) T*2T - T*T2 + 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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