Operator Algebras Generated by Left Invertibles

UNL 2019
University of Nebraska, Lincoln, NE

Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. Representations of such algebras encode the dynamics of orthonormal sets in a Hilbert space. We instigate a research program on concrete operator algebras that model the dynamics of Hilbert space frames.

The primary object of this thesis is the norm-closed operator algebra generated by a left invertible T together with its Moore-Penrose inverse. Of particular interest is when T satisfies a nondegeneracy condition called analytic. We show that T is analytic if and only if its adjoint belongs to an important class of operators called Cowen-Douglas. When the Fredholm index of T is -1, the algebra contains the compact operators, and any two such algebras are boundedly isomorphic if and only if they are similar. A classification of these algebras by K-theoritc means is then acquired.