A general Pietsch Domination Theorem

In this short communication we show that the unified Pietsch Domination Theorem proved in Botelho et al. (2010) [1] remains true even if we remove two of its hypotheses.     

On almost summing polynomials and multilinear mappings

In this article we explore the notion of everywhere almost summing polynomials and define a natural norm which makes this class a Banach polynomial ideal which is a holomorphy type and also coherent and compatible with the notion of almost summing linear operators. Similar results are not valid for the original concept of almost summing polynomials, due to G. Botelho.     

A note on multiple summing operators and applications

We prove a new result on multiple summing operators and, among other results and applications, we provide a new extension of Littlewood’s 4 / 3 inequality to m-linear forms.     

Lorentz summing mappings

In this paper we introduce and investigate a nonlinear concept of Lorentz summing operators. Some examples, counterexamples and connections with the theory of absolutely summing operators are presented (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Pietsch's factorization theorem for dominated polynomials

We prove that, like in the linear case, there is a canonical prototype of a p-dominated homogeneous polynomial through which every p-dominated polynomial between Banach spaces factors.     

Multiple summability properties for Cohen ‐summing operators

We prove that Cohen p‐summing operators satisfy multiple summability properties. Some of these multiple summability properties are new even in the linear case. For example, we prove that the multilinear functional associated to a Cohen p‐summing n‐linear operator is multiple ‐summing.     

Abstract extrapolation theorems for absolutely summing nonlinear operators

In this paper we prove a general version of the extrapolation theorem for absolutely summing nonlinear operators. It is explicitly shown that this result encompasses the previous old and recent, linear and nonlinear extrapolation theorems as particular cases. One of the steps of the proof provides another nonlinear extrapolation theorem of independent interest.     

Absolutely summing polynomials on Banach spaces with unconditional basis

If X is a Banach space with a normalized unconditional Schauder basis (xn), we define whenever and obtain estimates for μX,(xn) when every continuous m-homogeneous polynomial from X into Y is absolutely (q,1) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m=1, we obtain new contributions to the linear theory of absolutely summing operators.     

A factorization method for q-Racah polynomials

We develop a factorization method for q-Racah polynomials. It is inspired by the approach to q-Hahn polynomials based on the q-Johnson scheme, but we do not use association scheme theory nor Gel'fand pairs but only manipulation of q-difference operators.     

Around operators not increasing the degree of polynomials

We present a generic operator J defined on the vectorial space of polynomial functions and we address the problem of finding the polynomial sequences that coincide with the (normalized) J-image of themselves. The technique developed assembles different types of operators and initiates with a transposition of the problem to the dual space. We provide examples for a J limited to three terms.     

The correlation functions of vertex operators and Macdonald polynomials

The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n = 1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.     

Yosida-Hewitt type decompositions for weakly compact operators and operator-valued measures

We derive Yosida-Hewitt type decompositions for weakly compact operators from Köthe-Bochner function spaces to Banach spaces. As an application, we obtain a Yosida-Hewitt type decomposition for strongly bounded operator-valued measures.     

Ladder operators for Szego polynomials and related biorthogonal rational functions

We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szego and for their four parameter generalization to biorthogonal rational functions on the unit circle.

     

On the continuous analog of Rakhmanov's Theorem for orthogonal polynomials

We obtain the continuous analogs of Rakhmanov's Theorem for polynomials orthogonal on the unit circle. Sturm-Liouville operators and Krein systems are considered. For a Sturm-Liouville operator with bounded potential q, we prove the following statement. If the essential spectrum and absolutely continuous component of the spectral measure fill the whole positive half-line, then q decays at infinity in the certain integral sense.     

Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P_λ^(1/g) (χ₁, …, χ_n) …, χ_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_zk we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P_λ^(1/g) (x₁, …, x_k, 1, … 1). Using the operator Q_z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.