On the Permutability of Self-Adjoint Operators

Abstract

1. The permutability of two bounded self-adjoint operators T1 and T2 defined on a Hilbert space S& is defined in the most natural and obvious manner: We say that T1 commutes with T2 if T17T2 = T2T1 throughout S. If {E1(X) } and {E2(0-) } are the canonical resolutions of the identity of T7 and T2 respectively, then the following result is well known: A necessary and sufficient condition that T7 commutes with T2 is the validity of any one of the following conditions: (a) The product T1T2 or T2T1 is self-adjoint. (b) E1(X) commutes with E2(0-) for each X and o. If Ti is a bounded self-adjoint operator and T2 any (not necessarily bounded) self-adjoint operator, then one usually defines permutability as follows (cf. [2] p. 31 or [4] p. 404): T1 commutes with T2 if Ti T2 (7 T2T1 . That is, if and only if, whenever f is in the domain of T2, Tif is also in the domain of T2 and T71T2f = T2T1f. This definition is equivalent to (b). Therefore, the permutability of any two selfadjoint operators is usually defined by (cf. [2], p. 50). DEFINITION 1. Two self-adjoint operators T1 and T2 with the corresponding canonical resolutions of the identity {E1(X)} and {E2(0-) } respectively are said to be permutable (or T1 commutes with T2) if E1(X) commutes with E2(o-) for each X and a-. The objective of this note is to show that two self-adjoint operators T1 and T2 commute if there exists a self-adjoint operator T such that T ( 7 T1T2 or T a T2T71. For the standard results and techniques of general Hilbert space theory used in this paper the reader is referred to the various papers of J. von Neumann [3], [4] and the books by M. H. Stone [5] and B. v. Sz. Nagy [2]. 2. To elaborate on our statements in ?1 we first recall the definition of multiplication of unbounded operators. The product AB of two operators A and B is defined as follows: (AB)f is defined if and only if Bf and A (Bf) are both defined and is then equal to the latter. DEFINITION 2. We say that the operators S, T are C' if S and T are self-adjoint, D(T) z D(S) (D((T) denotes the domain of T) and for f, g E D((T), the inner product (Sf, Tg) is Hermitean symmetric in f and g. LEMMA 1. Suppose that S, T are C' and E is a projection operator which commutes with T and such that R(E) ( D((T) (R(E) denotes the range of E). Let Si and T1 be the restrictions of ES and T respectively to R(E). Then S1 and T1 are bounded self-adjoint operators and S1 commutes with T1 . PROOF. Let f, g E R(E). Since R(E) C D((T) C D(S), f and g belong to D(S) and (I-E)Sf and (I-E)Sg belong to S& E R(E). Therefore,