Abstract. The norm closure of the intermediate orbit of an essentially normal operator acting on an infinite dimensional separable Hilbert space need not contain any normal operator. Two instances of essentially normal operators where the norm closure of the intermediate orbit contains a normal operator are discussed.

AMS Subject Classification (1991): 47A58

Received August 24, 1992 and in revised form October 20, 1993. (Registered under 5574/2009.)