Normal extensions of formally normal operators

Abstract

However, until recently, conditions under which a formally normal operator N can be extended to a normal one in > were known only for certain special cases. 3 ' 4 Kilpi 5 considered the problem in terms of the real and imaginary parts of N. It is the purpose of this note to characterize the normal extensions of N in a manner similar to the von Neumann solution for the symmetric case. If Ni is a normal extension of a formally normal operator N in >, then it is easy to see that NaN x a N*, and NaN? <zN*. In Theorem 1 we describe (iV*) and (*) for any two operators N, N satisfying N c N*, iVciV*. With the aid of this result a characterization of the normal extensions i V of a formally normal N in is given in Theorem 2. It is indicated in Theorem 3 how the domains of normal extensions