A note on convolution in Banach spaces (Q762722)

Let X be a Banach space, and let B(X,X) denote the Banach algebra of all bounded linear operators from X into X. It is known that the reflexivity of B(X,X) implies the reflexivity of X. The converse of this result is in general false. Kalton (1974) has shown that if X is reflexive and non- separable then B(X,X) is not reflexive. Again if X is reflexive, and has the approximation property then B(X,X) is not reflexive (Holub 1973). In this paper the authors prove the following result. Write X' and X'' for respectively the first and second normed conjugate spaces of X. For \(F\in X'\), and for \(M\in X''\), let \(\bar MF\) be the continuous linear functional on B(X,X) defined for each g in B(X,X) by \((\bar MF)(g)=M(gF),\) where gF is the element of X' given by \((gF)(x)=F(g(x))\) \((x\in X)\). Then the reflexivity of X will imply that of B(X,X) provided that the linear span of the set of functionals \([\bar MF: M\in X''\), \(F\in X']\) is dense in the Banach space of all continuous linear functionals on B(X,X). This result is proved by applying the properties of a certain convolution operation in Banach spaces. This operation, described here by the authors, generalises in a wide sense the classical notion of convolutions of functions and measures. The work of Enflo (1973) shows the existence of separable reflexive Banach spaces without the approximation property. The authors' result will have significance if it can be shown that any of such spaces satisfy their density condition.