Essentially Hermitian Operators in B(L p )

Abstract

It is shown that on Lp[O, 1] all bounded linear operators which are in the Calkin algebra B(LP)/C(4), must be of the form Hermitian plus That is, essentially operators have the form, real multiplier plus compact. 1. Let X denote an infinite dimensional complex Banach space and B(X) the corresponding space of bounded (resp. compact) linear operators on X. The Calkin algebra associated with X is given by A(X) = B(X)/C(X). Many papers recently have dealt with variations of the following lifting question: Given that a coset T + C(X) in A (X) has a certain property, does the coset to an operator T + K, K E C(X), having the same property? For example, Stampfli [8] has shown that, if X is a separable complex Hilbert space, for every operator T E B(X) there is a compact operator KT so that the Weyl spectrum of T and the spectrum of T + KT are equal. In fact for most lifting theorems X is a separable infinite dimensional complex Hilbert space. Recently however, in an attempt to consider more general Banach spaces, these authors have proved that if X = Ip, then elements in the Calkin algebra lift to the form Hermitian plus In this paper the above result is extended to the case X = Lp[O, 1] (hereafter referred to as L.): namely, the essentially operators on B(Lp), 1 1, 4' is unique.) So given A,, 4'(t) = sgn 4'(t) I4(t) IP where sgn it = e-is if it = pei9. Finally, if 4 and 4' are unit vectors in L. then 4;8'(Qk>) = = f (t)4'(t) dt. We begin with a result proved in [1, Lemma 1], for T E B(Ql). The proof carries over to B(4L) with only one minor change. Whereas in the proof of [1] there existed a projection P E 6Y and unit vectors + and 4' in Lp for which = supp e q PTP-' , due to the fact that T and either P or P' were compact operators, here, it may only be assumed that for given E > 0, there exists a This content downloaded from 157.55.39.205 on Sun, 04 Dec 2016 04:55:31 UTC All use subject to http://about.jstor.org/terms ESSENTIALLY HERMITIAN OPERATORS 73 projection P E 'P and unit vectors 4 and 4' satisfying + > supp pE IPTP'Ilp. Nevertheless the following result still holds. LEMMA 1. Let T E B(LP), 1 < p < xo,p #'2. Then sup IIPTP'll < cpri(T) < so