Spectral multiplicity theory for a class of singular integral operators

Abstract

where x(A) is an arbitrary function of L2(a, b). The integral is to be interpreted as a Cauchy principal value, and f(A), are to be real valued, of class C2 on [a, b], such that the functions f '(A{) ? k'(X) have only finitely many zeros, while k(A) > 0 almost everywhere. Further, at each zero of f'(G) ? k'(A), we require the corresponding second derivative, f(k) ? k(A), to be nonzero (2). The contents of this paper have some points of contact with recent work of J. Schwartz [6], in which he analyzes the extent of the essential spectrum of general singular integral operators by using techniques of the theory of commutative Banach algebras. We consider only a special class of these operators, but our methods, which are fairly elementary in nature, allow us to obtain a complete spectral representation in this case. The important observation of Dr. Pincus that the hypothesis 6.1 of [1] is universally true motivated my search for a replacement of the lengthy computations of that article, and the main purpose of the discussion here is to give a concise argument leading to the main results which we now state(3).