Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

Domination spaces and factorization of linear and multilinear summing operators

It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1,?. . .?, pn)-dominated multilinear operators and dominated (p1,?. . .?, pn; σ)-continuous multilinear operators.     

A unified Pietsch domination theorem

In this paper we prove an abstract version of Pietsch's domination theorem which unify a number of known Pietsch-type domination theorems for classes of mappings that generalize the ideal of absolutely p-summing linear operators. A final result shows that Pietsch-type dominations are totally free from algebraic conditions, such as linearity, multilinearity, etc.     

Domination problems for strictly singular operators and other related classes

We survey recent results on domination properties of strictly singular operators and related operator ideals, as well as Banach–Saks operators, Narrow operators and p-summing operators.     

Operator Ideals,Orthonormal Systems and Lacunary Sets

Given an orthonormal system B in some L²(u) we consider the operator ideals II_B and T_B of B-summing and B-type operators and some related ideals. We characterize by certain weak compactness properties when II_B is equal to the operator ideal II₂ of 2-summing operators. In lose that B consists of characters of a compact abelian group we characterize when II_B coincides with the operator ideal II_γ of Gauss-summing operators and when T_B coincides with the operator ideal II_p of type-2 operators. Moreover, we give a necessary and sufficient condition for Fig to contain the operator ideal II_p of p-summing operators (2 < p < ∞) and for T_B to contain the operator ideal Γ_p of p - factorable operators.     

Multiple summing,dominated and summing operators on a product of l_1 spaces

We use the Maurey-Rosenthal factorization theorem to give a characterization of multiple 2-summing, 2-dominated and 2-summing operators (resp. multiple 1-summing, 1-dominated and 1-summing operators) on a product of \(l_{1}\) spaces, which extend a result of A. Nahoum in the linear case.     

Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences

In this work we study mappings f from an open subset A of a Banach space E into another Banach space F such that, once a∈A is fixed, for mixed (s;q)-summable sequences of elements of a fixed neighborhood of 0 in E, the sequence is absolutely p-summable in F. In this case we say that f is (p;m(s;q))-summing at a. Since for s=q the mixed (s;q)-summable sequences are the weakly absolutely q-summable sequences, the (p;m(q;q))-summing mappings at a are absolutely (p;q)-summing mappings at a. The nonlinear absolutely summing mappings were studied by Matos (see [Math. Nachr. 258 (2003) 71-89]) in a recent paper, where one can also find the historical background for the theory of these mappings. When s=+∞, the mixed (∞,q)-summable sequences are the absolutely q-summable sequences. Hence the (p;m(∞;q))-summing mappings at a are the regularly (p;q)-summing mappings at a. These mappings were also studied in [Math. Nachr. 258 (2003) 71-89] and they were important to give a nice characterization of the absolutely (p;q)-summing mappings at a. We show that for q<s<+∞ the space of the (p;m(s;q))-summing mappings at a are different from the spaces of the absolutely (p;q)-summing mappings at a and different from the spaces of regularly (p;q)-summing mappings at a. We prove a version of the Dvoretzky-Rogers theorem for n-homogeneous polynomials that are (p;m(s;q))-summing at each point of E. We also show that the sequence of the spaces of such n-homogeneous polynomials, n∈N, gives a holomorphy type in the sense of Nachbin. For linear mappings we prove a theorem that gives another characterization of (s;q)-mixing operators in terms of quotients of certain operators ideals.     

Completely Bounded and Ideal Norms of Multiplication Operators and Schur Multipliers

We estimate the completely bounded norms, the completely p-nuclear norms, and the completely p-summing norms of certain multiplication operators and Schur multipliers.     

Maurey’s factorization theory for operator spaces

We prove an operator space version of Maurey’s theorem, which claims that every absolutely (p, 1)-summing map on C(K) is automatically absolutely q-summing for q > p. Our results imply in particular that every completely bounded map from B(H) with values in Pisier’s operator space OH is completely p-summing for p > 2. This fails for p = 2. As applications, we obtain eigenvalue estimates for translation invariant maps defined on the von Neumann algebra V N(G) associated with a discrete group G. We also develop a notion of cotype which is compatible with factorization results on noncommutative L _p spaces.     

Weakly Compact and Absolutely Summing Polynomials

This paper shows that, contrary to the case of linear operators, absolutely summing homogeneous polynomials are not always weakly compact. It is also shown that, regardless of the infinite dimensional Banach space E and the positive integer n, there exists an n-homogeneous polynomial P from E to E that plays the role of the identity operator in the sense that P is neither compact nor absolutely r-summing for any r, and P is weakly compact if and only if E is reflexive.     

A Composition Theorem for Multiple Summing Operators

We prove that the composition S(u₁, …, u_n) of a multilinear multiple 2-summing operator S with 2-summing linear operators u_j is nuclear, generalizing a linear result of Grothendieck.     

Multiple summing operators on<Emphasis Type="Italic">C(K)</Emphasis> spaces

In this paper, we characterize, for 1≤p<∞, the multiple (p, 1)-summing multilinear operators on the product ofC(K) spaces in terms of their representing polymeasures. As consequences, we obtain a new characterization of (p, 1)-summing linear operators onC(K) in terms of their representing measures and a new multilinear characterization ofL _∞ spaces. We also solve a problem stated by M.S. Ramanujan and E. Schock, improve a result of H. P. Rosenthal and S. J. Szarek, and give new results about polymeasures. Both authors were partially supported by DGICYT grant BMF2001-1284.     

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch?s (p,r,s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q,p)-dominated operators.     

Cohen and multiple Cohen strongly summing multilinear operators

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.     

Interpolation of operator ideals with an application to eigenvalue distribution problems

We derive results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and thes-number idealsS _p ^s defined by Pietsch. By estimating theK-Functional of Peetre, we get that the interpolation ideal (S _p1 ^s ,S _p2 ^s ),_,p is contained inS _p ^s and is even equal to it in the case of the approximation numbers. A similar fact is proved for absolutely (p, q)-summing operators, interpolating the first index. We show further that the absolutelyp-summing operators onc ₀ are contained in the complex interpolation space ( _p1 (c _o), _p2 (c _o))_[].The previous results are then applied to prove summability properties for the eigenvalues of operators in Banach spaces, which are products ofS _p1 ^s -type and absolutelyp _j -summing operators. Roughly speaking, the summability order is the harmonic sum of thep _i - andp _j -indices, wherep _j 2. In the case of Hilbert spaces, this reduces to the well-known Weyl-inequality. The method uses an abstract interpolation estimate for ideal quasinorms which may be useful also for other operator ideals.     

Containment of c<Subscript>0</Subscript> and ℓ<Subscript>1</Subscript> in π<Subscript>1</Subscript>(<Emphasis Type="Italic">E,F</Emphasis>)

Suppose π₁(E, F) is the space of all absolutely 1-summing operators between two Banach spacesE andF. We show that ifF has a copy of c₀, then π₁ (E, F) will have a copy of c₀, and under some conditions ifE has a copy of ℓ₁ then π₁ (E, F) would have a complemented copy of ℓ₁.     

Multiple summing operators on Banach spaces

In this paper, we improve some previous results about multiple p-summing multilinear operators by showing that every multilinear form from spaces is multiple p-summing for 1?p?2. The proof is based on the existence of a predual for the Banach space of multiple p-summing multilinear forms. We also show the failure of the inclusion theorem in this class of operators and improve some results of Y. Meléndez and A. Tonge about dominated multilinear operators.     

Besov Spaces and a Trace Ideal

Let 2 be the operator ideal of all absolutely 2-summing operators and let (2)2,1(a) be the ideal of operators of 2-approximation type l2,1. In this paper we give some sufficient conditions for matrices and kernels related to Besov spaces to generate operators belonging to the ideal (2)2,1(a).     

A Composition Theorem for Multiple Summing Operators

We prove that the composition S(u₁, …, u_n) of a multilinear multiple 2-summing operator S with 2-summing linear operators u_j is nuclear, generalizing a linear result of Grothendieck. Both authors were partially supported by DGICYT grant BMF2001-1284.