Denseness of operators whose second adjoints attain their numerical radii

Abstract

We show that for any Banach space the set of (bounded linear) operators whose second adjoints attain their numerical radii is norm-dense in the space of all operators. In particular, the numerical radius attaining operators on a reflexive space are dense. B. Sims, paralleling the investigations by J. Lindenstrauss on norm attaining operators, raised the question of the norm denseness of numerical radius attaining operators. To date, only partial answers to this question have been given. Berg and Sims [1] got an affirmative answer for uniformly convex spaces. The same result was shown by C. S. Cardassi [3] for uniformly smooth spaces. Cardassi has also shown that numerical radius attaining operators on c0, 11, C(K) and L1(pu) are dense [4, 5, 6]. In this paper we obtain another sufficient condition for the denseness of numerical radius attaining operators, namely reflexivity of the space. In fact, we show that, given an operator T on a Banach space X, there exists a compact operator A, with arbitrarily small norm, such that the second adjoint of T + A attains its numerical radius. This result is analogous to the one obtained by Lindenstrauss for norm attaining operators [7, Theorem 1]. Our proof essentially consists of an adaptation of Lindenstrauss's proof in order to make it work for numerical radius attaining operators. The dual space of a normed space X will be denoted by X* and BL(X) will be the normed space of bounded linear operators on X. The numerical radius of such an operator T is defined by v(T) = Sup{ If( T(x))1: (x, f) E FI(X)} where In(X) = {(x ,f) E X x X*:f(x) = Ilf l = IIXHI = 1} and we say that T attains its numerical radius when there exists (x0, fo) in 11(X) such that Ifo(T(xo))I = v(T). Received by the editors March 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 47A 12; Secondary 46B 10.