Operator Algebras

Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. The Toeplitz algebra is a classic example. The generator $M_z$ on the Hardy space $H^2(\mathbb{T})$ is isometric, and hence, left invertible. Motivated by questions from linear equations in Hilbert spaces (frame theory), we wish to understand particular types of operator algebras generated by left invertible operators. In this talk, we investigate the norm closed operator algebra generated by a single left invertible operator $T$ in $\mathscr{B}(\mathscr{H})$ with a canonical left inverse (the Moore-Penrose inverse).

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.

This work focuses on operator algebras that arise from directed graphs and frame theory.