Abstract. Let $A$ and $B$ be selfadjoint operators on a Hilbert space with spectral families $E(\lambda )$ and $F(\lambda )$, respectively. $A\preceq B$ means $E(\lambda )\geq F(\lambda )$ for every $\lambda $, and this order is called the \it spectral order. \rm A selfadjoint operator valued real analytic function which is defined on $\bf R$ is called a \it real analytic wave \rm if it is piecewise monotone in the spectral order sense. We show that $ A\preceq A +s B$ for every $s>0$ implies $AB=BA$, and that a finite set of selfadjoint operators is commutative if and only if these operators are connected by a real analytic wave.

AMS Subject Classification (1991): 47B15

Received November 20, 1995. (Registered under 5724/2009.)