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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.
DeSantis, Derek, "Operator Algebras Generated by Left Invertibles" (2019). ETD collection for University of Nebraska-Lincoln. AAI13860332.