Szegö Type Limit Theorems

Abstract

Let S 1 be a unit circle with the standard measure, P k be the orthogonal projection in L 2 (S 1 ) on the subspace spanned by e ijx , j=0, \1, ..., \k, and B be the operator of multiplication by a smooth function b in L 2 (S 1 ). The classical Szego limit theorem states that under some assumptions on bThe operators P k coincide with the spectral projections P * of the selfadjoint operator (&d 2 Âdx 2 ) 1Â2 in L 2 (S 1 ) corresponding to the intervals [0, *) with k<* k+1. Following V. Guillemin [G], we obtain a generalization of this theorem for P * being the spectral projections of an elliptic selfadjoint (pseudo)differential operator A on a manifold without boundary. We also study the case where A is the operator of an elliptic boundary value problem.Moreover, we consider an arbitrary sufficiently smooth function instead of the logarithm. In other words, we obtain asymptotics of the functional

Spectral Enclosures and Complex Resonances for General Self-Adjoint Operators

“…They used their new formula in [23] to provide a generalization of the classical Szegő limit theorem, described below. Let P λ be the spectral projections of an auxiliary self-adjoint operator A, so that P λ converge strongly to the identity operator as λ → ∞.…”

“…Laptev and Safarov [23] obtained an asymptotic formula for this with an explicit error estimate, generalizing Szegő's theorem, and provided several applications to pseudodifferential operators. They assumed that the operators A and H satisfy certain commutator estimates, in order to obtain corresponding commutator estimates for P and H .…”

On the approximation of spectra of linear operators on Hilbert spaces

“…Let also W 2 ∞ denote the Sobolev space of measurable functions on R with second derivative (in the distributional sense) being L ∞ . [26].) Let A be a self-adjoint, unbounded operator on H and let P be projection such that P A is a Hilbert-Schmidt operator.…”

On the relation between an operator and its self-commutator

“…Denote its spectral projections corresponding to the intervals (0, λ) by P λ , and let A direct calculation shows that (G − λI ) −1 P λ−1 2 2 π 2 6 N 1 (λ) (see [19] for details). This estimate, (3.6) and the obvious inequality P λ − P λ−1…”