Daniel Pellegrino

Linear subsets of nonlinear sets in topological vector spaces

For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.

Lower bounds for the constants in the Bohnenblust–Hille inequality: The case of real scalars

The Bohnenblust-Hille inequality was obtained in 1931 and ( in the case of real scalars) asserts that for every positive integer m there is a constant Cm so that ((N)Sigma(i1 , . . . , im=1)vertical bar T(e(i1) (,...,) e(im))vertical bar(2m/m+1))(m+1/2) <= C-m parallel to T parallel to for all positive integers N and every m-linear mapping T : l(infinity)(N) x...x l(infinity)(N) -> R. Since then, several authors have obtained upper estimates for the values of C-m. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for C-m.

Inclusions and coincidences for multiple summing multilinear mappings

Using complex interpolation we prove new inclusion and coincidence theorems for multiple (fully) summing multilinear and holomorphic mappings. Among several other results we show that continuous n- linear forms on cotype 2 spaces are multiple (2; q(k),..., q(k))-summing, where 2(k-1) = 0.

Two new properties of ideals of polynomials and applications

Abstract In this paper we introduce two properties for ideals of polynomials between Banach spaces and showhow useful they are to deal with several a priori different problems. By investigating these properties we obtain, among other results, new polynomial characterizations of L∞ spaces and characterizations of Banach spaces whose duals are isomorphic to f 1 (Λ).

On the upper bounds for the constants of the Hardy–Littlewood inequality☆

Abstract The best known upper estimates for the constants of the Hardy–Littlewood inequality for m-linear forms on l p spaces are of the form ( 2 ) m − 1 . We present better estimates which depend on p and m. An interesting consequence is that if p ≥ m 2 then the constants have a subpolynomial growth as m tends to infinity.

Absolutely summing multilinear operators: a panorama

Abstract This paper has a twofold purpose: to present an overview of the theory of absolutely summing operators and its different generalizations for the multilinear setting, and to sketch the beginning of a research project related to an objective search of “perfect” multilinear extensions of the ideal of absolutely summing operators. The final sections contain some open problems that may indicate lines for future investigation.

Towards sharp Bohnenblust–Hille constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust--Hille type inequalities for

Dominated polynomials on infinite dimensional spaces

m

On almost summing polynomials and multilinear mappings

--linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust--Hille inequality are universally bounded, irrespectively of the value of

Estimates for the asymptotic behaviour of the constants in the Bohnenblust–Hille inequality

m