On the Permutability of Normal Operators

Abstract

1. In this paper we obtain a theorem about the permutability of (in general unbounded) normal operators. This theorem is analogous to a theorem proved by the present authors in collaboration with J. von Neumann [3] concerning the permutability of self-adjoint operators. As we shall point out later it is not possible to generalize precisely the conditions given in [3] to the case of normal operators. Nevertheless, the proof of the theorem given here, with essential modifications, follows the proof given in [3]. We give our theorem and proof in Section 2. In Section 3 we apply this theorem on the permutability of normal operators to obtain the integral representation for a semi-group {N,} of normal operators (in general unbounded), where the index x ranges over a semi-group which is a cartesian product I+ X E where I+ is the set of vectors in m-dimensional euclidean space with non-negative integer components and Et is the set of vectors in n-dimensional euclidean space with non-negative components. This theorem contains as special cases theorems by E. Hille [6], B. v.Sz. Nagy [9, pp. 74-76], A. Devinatz [2] and a theorem recently published by R. Getoor [5]. It should be pointed out that our theorem represents an essential generalization of these previous results. For the case where { Nx ; x > 0 } is a one parameter semi-group of unbounded normal operators, the method of Nagy [9, pp. 74-75] may be pushed through to obtain the integral representation, and only in this case. However, with the use of our permutability theorem we are able to materially simplify the proof for the one parameter case. Moreover, in the unbounded case, even with the use of our permutability theorem, there remain some difficulties for multi-parameter semi-groups which do not allow us to use immediately the theorem for the one-parameter case. If the index x ranges over a more general Abelian topological semi-group than we consider, the representation problem is still open, although for semi-groups of normal operators which are uniformly bounded a method introduced by R. Phillips [8] may be used (cf. also A. E. Nussbaum [7]). 2. We start with the definition of permutability of two (in general unbounded) normal operators defined in a Hilbert space & (cf. Nagy [9] p. 50). DEFINITION 1. Let N1 and N2 be normal operators and {K1(z)} and {K2(Z)I their corresponding canonical resolutions of the identity. We say that N1 and N2 permute if Kl(zi)K2(z2) = K2(z2)Kl(zi) for each complex z1 and z2 It should be pointed out here that this definition is, by a theorem of B. Fuglede [4], equivalent to the usual definition of permutability in the case where N1 and N2 are bounded (N1N2 = N2N1) or in the case where N1 is bounded and N2 is unbounded (N1N2 _ N2N1). DEFINITION 2. We say that the operators N1, N2 and N have the property P if they are normal operators and if N = N1N2 = N2N1 .