Tensor products and Banach ideals of p-compact operators

A study is made of the norm w_p (1 p ) on the tensor product of two Banach spaces E and F. It is shown that w_p is a tensor norm, and a representation is deduced for the elements in the completion of E F equipped with w_p. Finally it is shown that the w_p-nuclear operators in the sense of Grothendieck [3] coincide with those operators factoring compactly through^p (if 1 p ) or Co (if p=), with related equalities concerning the idea1 norms.     

Banach ideals of operators with applications

We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the $\text{L}$_p-spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in L_p(μ)-spaces.     

Banach ideals of operators with applications to the finite dimensional structure of Banach spaces

We present a few applications of the theory of Banach ideals of operators. In particular, we give operator characterizations of the ℒ _p spaces, compute the relative projection constant of isometric embeddings of Hilbert spaces inL _p -spaces, and show that Π₁ (E, F), the space of absolutely summing operators, is reflexive ifE andF are reflexive andE has the approximation property. Research supported by NSF-GP-34193 Research supported by NSF-Science Development Grant     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

Hankel operators in Schatten ideals

We study membership to Schatten ideals S _E , associated with a monotone Riesz–Fischer space E, for the Hankel operators H _f defined on the Hardy space H ²(∂D). The conditions are expressed in terms of regularity of its symbol: we prove that H _f ∈S _E if and only if f∈B _E , the Besov space associated with a monotone Riesz–Fischer space E(dλ) over the measure space (D,dλ) and the main tool is the interpolation of operators. Received: December 17, 1999; in final form: September 25, 2000?Published online: July 13, 2001     

Unconditional ideals of finite rank operators

Let X be a Banach space. We give characterizations of when is a u-ideal in for every Banach space Y in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when is a u-ideal in for every Banach space Y, when is a u-ideal in for every Banach space Y, and when is a u-ideal in for every Banach space Y.     

Maximal ideals of disjointness preserving operators

Let L and M be vector lattices with M Dedekind complete, and let Lr(L,M) be the vector lattice of all regular operators from L into M. We introduce the notion of maximal order ideals of disjointness preserving operators in Lr(L,M) (briefly, maximal δ-ideals of Lr(L,M)) as a generalization of the classical concept of orthomorphisms and we investigate some aspects of this ‘new’ structure. In this regard, various standard facts on orthomorphisms are extended to maximal δ-ideals. For instance, surprisingly enough, we prove that any maximal δ-ideal of Lr(L,M) is a vector lattice copy of M, when L, in addition, has an order unit. Moreover, we pay a special attention to maximal δ-ideals on continuous function spaces. As an application, we furnish a characterization of lattice bimorphisms on such spaces in terms of weigthed composition operators.     

Relatively regular ideals in semiprime Banach algebras

Nella prima parte di questo lavoro vengono studiate le relazioni che intercorrono tra il concetto di relativa regolarità in un'algebra non necessariamente unitaria ed alcuni ideali (radicale, zoccolo, prezoccolo) che intervengono nello sviluppo della teoria di Fredholm in algebre di Banach. Nella seconda parte del lavoro viene dimostrato che in certe algebre di BanachA complesse semiprime l'esistenza di un ideale relativamente regolare chiuso implica cheA è unitaria e di dimensione finita.     

On Banach lattices of operators

Let Λ₁ and Λ₂ be infinte-dimensional, Banach lattices such thatc _o is not finitely representable in Λ₂. Then the bounded linear operators from Λ₁ to Λ₂ form a lattice if and only if Λ₁ is an abstract AL space.     

Derivations into duals of ideals of Banach algebras

We introduce two notions of amenability for a Banach algebra A. LetI be a closed two-sided ideal inA, we sayA is I-weakly amenable if the first cohomology group ofA with coefficients in the dual space I* is zero; i.e.,H ¹(A, I*) = {0}, and,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study theI-weak amenability of a Banach algebraA for some special closed two-sided idealI.