Tensor Products and Grothendieck Type Inequalities of Operators in ¿„-spaces Bernd Carl and Andreas Defant

Several results in the theory of (p, £?)-summing operators are improved by a unified but elementary tensor product concept. Introduction Since the pioneering work of Grothendieck in Linear Functional Analysis there is an extensive literature dealing with operators in Lp-spaces. Still the most important result in this direction is Grothendieck's integral characterization [9] of operators from l\ into l2 : There is a universal constant Kg > 0 such that for every operator S : l\ —► l2 there is a probability measure v on the unit ball B¡ao of 4o (endowed with its weak* topology) for which \\Sx\\ < KG\\S\\ J \(x, a)\du(a) holds for all x e l\. For information on estimates of the constant KG we refer to [20]. This result which is now called Grothendiek's Theorem—Grothendieck himself called it "the fundamental theorem of the metric theory of tensor products"—motivated the following statement of this paper: there is an absolute constant k > 0 such that for every operator S : l\ —► lv ( 1 < v < oo) and every probability measure p on B¡v, (v' the conjugate index v/(v-l)) there is a probability measure v on B¡aa with (J\(Sx,a)\sdp(a)^ ' /„ (1 < v < oo) maps an unconditional summable sequence (x,) in l\ into a sequence (Sx¡) which is the product of an absolutely s'-summable scalar sequence (a,) and a weakly ssummable sequence (y,) in lv , here again 2 < s < oo and j = \j £ |. Within the theory of absolutely summing operators which was initiated by Pietsch [22], our result has the following formulation: If 2 < s < oo and } = | \ — £ |, Received by the editors March 6, 1989 and, in revised form, January 10, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B20, 47B10, 46M05. © 1992 American Mathematical Society 0002-9947/92 $1.00+ $.25 per page