Personal Notes in Operator Algebras and Operator Theory

This note collects some facts, theorems in operator algebras and operator theory for fullfilling my understanding in my studying progress. Part I contains some fundamental notes which I collect from many textbooks and surveys. I will either try to provide some unclear details or solve some basic but important exercises. Other parts might include some parts in published articles which I try to fill in more details. The choices of the topics here are due to my personal taste. Notes are very incomplete and contains certain errors. Comments are welcome. 1. Some fundamental facts in Operator Theory and Operator Algebras 1.1. Some preliminaries in spectral theory of bounded linear operators. We start with a simple statement about the relation of the support of the spectral measure and the spectrum of a normal operator T . Proposition 1.1. Let T ∈ B(H) be a normal operator. Then a complex number λ ∈ σ(T ) if and only if ET (U) 6= 0 for every open subset U containing λ. Here is an important statement concerning isolated points in spectrum of normal operators (Exercise 11, p.75 of [10]): Proposition 1.2. Let T be a normal operator in B(H) where H is a Hilbert space. Then if x is an isolated point of the spectrum σ(T ), we have x is an isolated eigenvalue of T and the spectral measure ET ({x}) is the projection onto the eigenspace Ker(T − xI). Moreover, T has a nontrivial invariant closed vector subspace if dim(H) > 1. Proof. The function χ{x}(z−x) is identical to the zero function on σ(T ). Hence (T − xI)ET ({x}) = 0 by continuous functional calculus and so ET ({x}) is a subprojection of Ker(T − xId). Since the subspace Ker(T − x) is an invariant subspace under T and T ∗, the projection P onto Ker(T − x) should be regarded as a characteristic function χE in L ∞(σ(T ), μ) for some E ∈ Bσ(T ) and so x ∈ E. Since (T − x)P = 0, χE(z)(z − x) = 0, μ− a.e z ∈ σ(T ) or μ{E \ {x}} = 0. This implies that χE = χ{x} or P = ET ({x}). Because x is isolated in σ(T ), it follows that ET ({x}) > 0 and this means the eigenspaceKer(T−x) is nontrivial or x is an eigenvalue for T . TET ({x}) = xET ({x}) yields that the range of ET ({x}) is a nontrivial invariant subspace for T in the case ET ({x}) < 1. Otherwise, T = xId and this case is totally trivial if dim(H) > 1. 2000 Mathematics Subject Classification. 20C07, (20E99).