Operator Algebras Generated by Left Invertibles

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 T†. We denote this algebra by AT. In the isometric case, T† = T* and AT is a representation of the Toeplitz algebra. Of particular interest is the case when T satisfies a non-degeneracy condition called analytic. We show that T is analytic if and only if T* is Cowen-Douglas. When T is analytic with Fredholm index –1, the algebra AT contains the compact operators, and any two such algebras are boundedly isomorphic if and only if they are similar