### Topics in Operator Theory, Lecture 1: Introduction

This is a summary of the first lecture, which was introductory in nature.

will always denote a Hilbert space over . will always denote the algebra of bounded operators on . I am interested in operators on Hilbert space; various subspaces and algebras of operators that come with various structures, as well as the relationship between these subspaces and structures; and connections and applications of the above to other areas, in particular complex function theory and matrix theory.

I expect students to know the spectral theorem for normal operators on Hilbert space (see here. A proof in the selfadjoint case that assumes very little from the reader can be found in my notes, see Section 3 and 4). I also will assume some familiarity with Banach algebras and commutative C*-algebras – the student should contact me for references.

We begin by surveying different kinds of structures of interest. Read the rest of this entry »