We invent the new notion of coordinatewise multiple summing operators in Banach spaces, and use it to study various vector valued extensions of the well-know Bohnenblust-Hille inequality (which originally extended Littlewood's 4/3-inequality). Our results have application on the summability of monomial coefficients of m-homogeneous polynomials P : ∞ → p , as well as for the convergence theory of products of vector valued Dirichlet series.

Multiple summing operators have been proven to be useful in several areas of analysis and mathematical physics. In this paper we prove reverse inclusions for this class of operators, completing the work already initiated in [D. Perez-Garcia, The inclusion theorem for multiple summing operators, Studia Math. 165 (3) (2004) 275-290].

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We give for some Banach spaces X and Y examples of linear and continuous operators U : C ( T , X ) + Y , such that U' cp E As,(X, Y ) , for each cp E C ( T ) and U # : C ( T ) + As,(X,Y) is a 2-absolutely summing operator with respect to the 2-absolute norm on As,(X, Y ) , but U is not 2-absolutely summing. o 1999 Aca deniic Press For X a Banach space, 1 I r I a, and ( x , , )~ c X we write wr(xn I n E N ) = sup{(C:=lIx*(x,,)I ) Ix* E X*, Ilx*ll I 11, if r < a and wx(xn I n E N ) = ~~p ,~~~l l x~l l , if r = m, w,(X> = { ( x ,~>~~~ cx I wr(x,, In E N )

Abstract:In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible -linear versions of Grothendieck's composition theorem in the case ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact. MSC:47H60, 46G25, 46B25, 46C99

We prove composition results for multilinear operators and multilinear variants of Maurey and Pietsch theorems both for multiple p-summing, p-dominated and p-summing multilinear operators.

Two inequalities resembling the multilinear Hölder inequality for mixednorm Lebesgue spaces are proved. When specialized to single-function inequalities they include a pair of inequalities due to Blei and a recent extension of Blei's inequality. The first of these inequalities is applied to give explicit indices in a known result for coordinatewise multiple summing operators. The second is used to prove a complementary result to the known one, again with explicit indices. As an application of the complementary result, a sufficient condition is given for a composition of operators to be multiple summing.

We prove that for some classes of weakly compact operators on the space C[0, 1] the conditions that they are compact, absolutely summing and nuclear are distinct and connected to vector-valued sequence spaces. Mathematics Subject Classification (2000). 46B28, 47B10, 47G10.