These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensio...

We define the concept of instability index of an isolated eigenvalue of a non-self-adjoint operator, and prove some of its general properties. We also describe a stable procedure for computing this index for Schrödinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential. AMS subject classification: 34L05, 35P05, 47A75, 49R...

Let Φ1, . . . ,Φn be strictly positive linear maps from a unital C∗algebra A into a C∗-algebra B and let Φ = ∑n i=1 Φi be unital. If f is an operator convex function on an interval J , then for every self-adjoint operator A ∈ A with spectrum contained in J , the following refinement of the Choi– Davis–Jensen inequality holds:

We prove that the spectrum of certain non-self-adjoint Schrödinger operators is unstable in the semi-classical limit h → 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semi-classical modes of the operator by the JWKB method for energies far from the spectrum.

We define a class of pseudo-ergodic non-self-adjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for su...

We consider an elliptic self-adjoint first-order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the...

We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein's trace formula and to the Birman-Solomyak spectral averaging formula are discussed.

Abstract Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper gives sufficient conditions for an integrable operator to be the square of a Hankel operator, and applies the condition to the Airy, associated Laguerre, modified Bessel and Whittaker func...

The main goal of the paper is to generalize the Duistermaat-Guillemin trace formula to the case of transversally elliptic operators on a compact foliated manifold. First, let us recall briefly the setting of the classical formula. Let P be a positive self-adjoint elliptic pseudodifferential operator of order one on a closed manifold M (for example, P = √ ∆, where ∆ is the Laplace-Beltrami opera...

In general, a symmetric operator does not have a unique spectral function. If T is a symmetric operator and F(t) is a spectral function for T, by the theory of Naimark (see Akhiezer and Glazman [1]) there exists a Hilbert space H of which H is a subspace and a self-adjoint operator T onif with resolution of the identity E(t) such that T + is an extension of T and F(t) = P£(<) where P is the pro...