Continuous functions of Hermitian operators

Abstract

Theorem: every operator is a function of a one. Corollary: every operator on a separable Hilbert space is the sum of a diagonal operator and a compact one. THEOREM. Every operator is a function of a one. COROLLARY. Every operator on a separable Hilbert space is the sum of a diagonal operator and a compact one. Any two commutative operators are functions of some operator; the version of the theorem in which continuous is replaced by Borel is old. Both the and the versions extend to countable sets of commutative operators, with no additional effort. The present version is an easy consequence of well known Hilbert space techniques, together with an elegant fact of general topology. The result, with a similar but different proof, is implicit in a construction of Schwartz [6, pp. 15-16]. Much is known about operators, but not everything. The theorem is presented here in the hope that it may yield some new information. The basis of the hope is the derivation of the corollary from the theorem. The version of the corollary in which normal is replaced by Hermitian is the Weyl-von Neumann theorem; till quite recently the extension to the case was an open problem [4]. The first solution is due to I. D. Berg [1]. The proof below is shorter, and it may perhaps be considered more translucent. PROOF OF THE THEOREM. Every compact metric space is a image of the Cantor set [5, ?41, VI]. If, in particular, A is a non-empty compact subset of the complex plane and 1F is the Cantor set in the unit interval, then there exists a function p from 17 onto A. The elegant fact from general topology enters here: it is the existence of a cross section. That is: there exists a function Vp from A into F such that the composition p o Vy is the identity on A [5, ?43, IX]. (Short Received by the editors February 3, 1971. AMS 1970 subject classifications. Primary 47B15. I Research supported in part by a grant from the National Science Foundation. (American Mathematical Society 1972