Szegö Type Limit Theorems

✍ Scribed by A. Laptev; Yu. Safarov

• Book ID: 102972293
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 720 KB
• Volume: 138
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0075

✦ Synopsis

Introduction

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 b

The 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 \ * ( )=Tr P * (P * BP * ) P * =: k (+ k (*)), (0.2) article no.