QUASI - SIMILAR MODELS FOR NILPOTENT OPERATORS ( x )

Every nilpotent operator on a complex Hilbert space is shown to be quasi-similar to a canonical Jordan model. Further, the para-reflexive operators are characterized generalizing a result of Deddens and Fillmore. A familiar result states that each nilpotent operator on a finite dimensional complex Hubert space is similar to its adjoint. One proof proceeds by showing that both a nilpotent operator and its adjoint have the same canonical form. In this note we show that although this result does not extend to infinite dimensional spaces, the weaker quasi-similarity version of it, together with the proof indicated above, still holds on any Hilbert space. This yields an affirmative answer to a question raised by P. Rosenthal in connection with the content of [3]. The canonical form exhibited provides positive evidence that the theory of Jordan models might be extended to cover operators of class C0 of infinite multipUcity and indeed, considerable progress [2] has been made recently in this direction. Although the Jordan model for nUpotent operators on infinite dimensional Hilbert spaces is no longer unique, we single out a "canonical" model. A similar result has been obtained independently by Berkovici [1]. We conclude with an application of our results to extend to infinite dimensional spaces a theorem of Deddens and Fülmore [4] which characterizes reflexive operators on finite dimensional spaces. We want to thank Lawrence WiUiams for pointing out an error in an earUer version of this note. 1. In this note, a nilpotent operator T will be caUed a Jordan operator if T = ®0lT0t, where each Ta operates on some C a for 0 < la < °° by the Jordan one-ceU matrix Received by the editors October 24, 1975. AMS (MOS) subject classifications (1970). Primary 47B99; Secondary 47A15, 47A45. (!) Research partially supported by a grant from the National Science Foundation. Copyright