Compact multilinear transformations

It is well known that if a C' map between Banach spaces is compact, then its derivative is a compact operator. If the map is Cr, then what can be said about the compactness of its higher derivatives? This question leads us to a study of compact multilinear operators with the main result being that the higher derivatives of a compact map are such operators. Let E1, * *, Em, F be Banach spaces, and T: El x xEm,--F be a continuous m-multilinear transformation. We call Tjointly compact if it takes bounded sets into relatively compact sets, and we call T separately compact if for eachj and any choice of points eicEi, i$j, the transformation T(el, * * *, e,_l, e,+,, + *, em):Ej->F is compact. The object of this paper is to expose several simple properties of such transformations. We give an example (?A) where separate compactness does not imply joint compactness, and we show that the first r derivatives of a compact C7 function are jointly compact (?B). We then look at linear spaces of jointly compact transformations (?C) and investigate their behavior with respect to tensor products (?D). (A) Let T:12xl2-42 be defined as T(a, b)=(albl, a2b2, ) where a= (a,, a2, * *) and b=(bl, b2, * * ) are in 12. Then T is bilinear, continuous, and separately compact but not jointly compact. Continuity follows from the inequalities I1T(a, b) I = ( a2b2)1/2 a)114(> b4) 1/4 (2 ai) '4(> b2)1/4 < I for 2 a2, , b2 < 1 To show separate compactness fix bel2 and take N so that M