Characterizations of new cohen summing bilinear operators

Abstract

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X x Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁, ..., X_n and all p-summing operators U : X₁ x ... x X_n → X, the operator V ◦ (U, I_Y ) : X₁ x ... x X_n x Y → Z is (1; p, ... , p, p*)- 

                                                                                       ︸ n-times                                                                                                

summing. The second result is: Let H be a Hilbert space, Y , Z Banach spaces and V : H x Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ◦ (U; I_Y) is (p, p*)-dominated.

Mathematics Subject Classification (2010): Primary 47H60, 46G25; Secondary 46B25, 46B28, 46B45, 46A32.

Keywords:  p-summing, dominated, multilinear operators, new Cohen summing bilinear operator