Singular Integral Operators on a Half-Line

Abstract

1. There is a well-perfected Fredholm theory for singular integral operators on Rn or compact manifolds, initiated principally by Calderon and Zygmund,' and Seeley,2 which has been extended to singular integrodifferential operators, or pseudodifferential operators, by various authors.3 The presence of boundary points, however, provides considerable additional difficulties for which there does not seem to exist an equally smooth treatment, in spite of strong activity and important progress during the last 2 years (Dynin,4 Agranovichj Vishik and Eskin,6 and others3). This paper develops a complete Fredholm theory for an algebra $3 of singular integral operators on the half-line R+ = [0, cx), which seems to display all the inherent simplicity of Noether's algebra' on R. 3 is a Banach algebra with osymbol in the sense of Breuer and Cordes.8 We show that the domain M of the symbol of an operator in $3 is homeomorphic to a circle, with a whole segment lying over the boundary points x = 0 and x = o each. Thus M is not a fiber bundle anymore. Our singular integral operators exhibit the same relationship to the Laplace operator-with all possible self-adjoint boundary conditions at x = 0-as in the theory of singular integral operators on a manifold without boundary. Extension to higher dimensional half-spaces, however, leads into theory of R-algebras,9 since the corresponding operators do not have compact commutators (cf. Lemma 1) and the generated algebra with a-symbol is to be taken with respect to an Ralgebra-ideal different from the compact ideal.Results and methods seem to be quite different from the work mentioned above, and we express the hope it might offer a different aspect and, perhaps, advantages to the known theory.Our proofs depend in part on the study of certain convolutionlike products which are related to the topological group (-1,1) under the group operation a (D i 1 + a To find the domain of the symbols of operators in $3 we employ the simple method used by Herman'0 for singular integral operators on R'. The notations (B-C), and (H), will refer to the above-mentioned papers of Breuer and Cordes, and Herman, respectively.2. Define the integral operators K+ and K-by

Gel'Fand Theory of Pseudo Differential Operators

“…Theorem 36 is proved by the same methods as applied for our earlier results concerning the algebra D. The proof is omitted. However, some details can be found in [15] since, when n 1, e is unitarily equivalent to the algebra studied there.…”

“…To this end, we bring to bear an approach in the theory of singular integral operators which originated with Goh'berg [5] and has since been elaborated and exploited by Seeley, Breuer, and the authors [1,2,3,6,10,15].…”

Does Levinson’s theorem count complex eigenvalues?

“…Summing up these information one gets : The content of Theorem 2 can now be deduced either from [9,Thm. 4.4] (with n = −1) or from [2,Thm. 3].…”

ON THE WAVE OPERATORS AND LEVINSON'S THEOREM FOR POTENTIAL SCATTERING IN ℝ3

“…has for all k ∈ R 3 with k 2 = 0 a unique solution with the second term in the r.h.s. of (11) in C 0 (R 3 ) for the x-variable. Furthermore, the following asymptotic development holds:…”