Banach Actions Preserving Unconditional Convergence

Let A,X,Y be Banach spaces and AxX & RARR;Y, (a,x)?ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n & ISIN;omega in A and unconditionally convergent series n-ary sumation n & ISIN;omega xn in X, the series n-ary sumation n & ISIN;omega anxn is unconditionally convergent in Y. We prove that a Banach action AxX & RARR;Y preserves unconditional convergence if and only if for any linear functional y*& ISIN;Y* the operator Dy*:X & RARR;A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from l1 to l2, we prove that a Banach action AxX & RARR;Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n & ISIN;omega such that for every x & ISIN;X, the series n-ary sumation n & ISIN;omega enx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r & ISIN;[1,& INFIN;] with 1r & LE;1p+1q, the coordinatewise multiplication lpxlq & RARR;lr preserves unconditional convergence if and only if one of the following conditions holds: (i) p & LE;2 and q & LE;r, (ii) 2