A Jordan Decomposition for Operators in Banach Space
Shmuel Kantorovitz · Transactions of the American Mathematical Society · 1965
Introduction.Let A" be a finite dimensional complex Euclidean space, and let T be a linear operator acting on X.The Jordan decomposition theorem states that T has a unique decomposition T= S + N, where S = jff(r)z d£(z), £ is a spectral measure supported by the spectrum o(T) of T, and N is a nilpotent operator commuting with S.Our main result (Theorem 2.1) is a generalization of the Jordan theorem for operators with real spectrum to infinite dimensional reflexive Banach spaces.We consider operators T satisfying the growth condition \e"T\ = 0(|i|*) for some integer fe^O and all real t.In §1, we construct the "Jordan manifold" for T, on which T is shown to have a unique Jordan decomposition, if the spectrum (which is real because of the growth condition) has Lebesgue measure zero ( §2).Related results are described in §2.The theory is illustrated by examples in §3.This work is clearly related to Dunford's theory of spectral operators.However, the latter is needed as a prerequisite only for Theorem 2.12.The standard reference is [1], [2].Many thanks are due to Professors H. Furstenberg and C. A. McCarthy for discovering an error in the original version of this paper.