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We show that the r-dominated polynomials on p (2 p ∞) are integral on 1 , and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on p (1 < p ∞) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.In the last years, many authors have studied r-dominated, integral, and nuclear polynomials on Banach spaces [1], [3], [4], [16], [17], [20]. More specifically, the r-dominated polynomials on p spaces have been treated in [17], and the nuclear polynomials on p have been analyzed in [3]. These two papers consider in particular the diagonal polynomials on p (the definitions are given below). We start by showing that the r-dominated polynomials on p (2 p ∞) are integral on 1 , proving that the converse is not true. Our counterexamples are 2-homogeneous, diagonal polynomials.Then we concentrate on the 2-homogeneous, diagonal polynomials on p (1 < p ∞), characterizing when they are r-dominated (the case p = 1 may be seen in [17, Theorem 2]). We show that, unlike the linear case, there are many nuclear polynomials which are not 1-dominated.Throughout, E and F denote Banach spaces, E * is the dual of E, and B E stands for its closed unit ball. By N we represent the set of all natural numbers, and K is the scalar field. We write L(E, F ) for the space of all (linear bounded) operators from E into F .We use the notation E ⊗ E for the two-fold tensor product of E, E ⊗ π E for the two-fold projective tensor product of E, and E ⊗ s E for the two-fold symmetric tensor product Mathematics Subject Classification (2000): Primary 46G25; Secondary 47H60.This work was performed during a visit of the first named author to the Universidad Politécnica de Madrid.The second named author was supported in part by Dirección General de Investigación, BFM 2003-06420 (Spain).

A well-known result due to W.B. Johnson (1971) asserts that the formal identity operator from 1 into ∞ is universal for the class of non-compact operators between Banach spaces. We show that there is neither a universal non-compact polynomial nor a universal non-unconditionally converging polynomial between Banach spaces.

We study the (DPrcp) and the Gelfand Phillips property in the space of compact operators. Moreover we give some sufficient conditions in order that a projective tensor product of two Banach spaces is sequentially Right (SR) or it has the L-limited property. Moreover we study the Bourgain Diestel property (BD), the (RDP *) property in the space K w * (X, Y). We introduce the dual (SR *) property and we give a characterization of it.

In this note we study the property (V) of Pelczynski, in a Banach space X, in relation with the presence, in the dual Banach space X * , of suitable weak * basic sequences. We answer negatively to a question posed by John and we prove that, if X is a Banach space with the Property (V) of Pelczynski and the Gelfand Phillips property, then X is reflexive if and only if every quotient with a basis is reflexive. Moreover, we prove that, if X is a Banach space with the property (V ) of Pelczynski, then either X is a Grothendieck space or W (X, Y ) is uncomplemented in L(X, Y ) provided that Y is a Banach space such that W (X, Y ) ̸ = L(X, Y ). (2010): Primary: 46M05. Mathematics Subject Classification