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Extension and lifting of weakly continuous polynomials

Raffaella Cilia, Joaquín M. Gutiérrez (2005)

Studia Mathematica

We show that a Banach space X is an ℒ₁-space (respectively, an -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space w b ( m X ) of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an -space.

Factorizing multilinear operators on Banach spaces, C*-algebras and JB*-triples

Carlos Palazuelos, Antonio M. Peralta, Ignacio Villanueva (2009)

Studia Mathematica

In recent papers, the Right and the Strong* topologies have been introduced and studied on general Banach spaces. We characterize different types of continuity for multilinear operators (joint, uniform, etc.) with respect to the above topologies. We also study the relations between them. Finally, in the last section, we relate the joint Strong*-to-norm continuity of a multilinear operator T defined on C*-algebras (respectively, JB*-triples) to C*-summability (respectively, JB*-triple-summability)....

Fully absolutely summing and Hilbert-Schmidt multilinear mappings.

Mário C. Matos (2003)

Collectanea Mathematica

The space of the fully absolutely (r;r1,...,rn)-summing n-linear mappings between Banach spaces is introduced along with a natural (quasi-)norm on it. If r,rk C [1,+infinite], k=1,...,n, this space is characterized as the topological dual of a space of virtually nuclear mappings. Other examples and properties are considered and a relationship with a topological tensor product is stablished. For Hilbert spaces and r = r1 = ... = rn C [2,+infinite[ this space is isomorphic to the space of the Hilbert-Schmidt...

Fully summing mappings between Banach spaces

Mário C. Matos, Daniel M. Pellegrino (2007)

Studia Mathematica

We introduce and investigate the non-n-linear concept of fully summing mappings; if n = 1 this concept coincides with the notion of nonlinear absolutely summing mappings and in this sense this article unifies these two theories. We also introduce a non-n-linear definition of Hilbert-Schmidt mappings and sketch connections between this concept and fully summing mappings.

Hilbert spaces of analytic functions of infinitely many variables

O. V. Lopushansky, A. V. Zagorodnyuk (2003)

Annales Polonici Mathematici

We study spaces of analytic functions generated by homogeneous polynomials from the dual space to the symmetric Hilbertian tensor product of a Hilbert space. In particular, we introduce an analogue of the classical Hardy space H² on the Hilbert unit ball and investigate spectral decomposition of unitary operators on this space. Also we prove a Wiener-type theorem for an algebra of analytic functions on the Hilbert unit ball.

Holomorphy types and ideals of multilinear mappings

G. Botelho, H.-A. Braunss, H. Junek, D. Pellegrino (2006)

Studia Mathematica

We explore a condition under which the ideal of polynomials generated by an ideal of multilinear mappings between Banach spaces is a global holomorphy type. After some examples and applications, this condition is studied in its own right. A final section provides applications to the ideals formed by multilinear mappings and polynomials which are absolutely (p;q)-summing at every point.

Holomorphy types and spaces of entire functions of bounded type on Banach spaces

Vinícius V. Fávaro, Ariosvaldo M. Jatobá (2009)

Czechoslovak Mathematical Journal

In this paper spaces of entire functions of Θ -holomorphy type of bounded type are introduced and results involving these spaces are proved. In particular, we “construct an algorithm” to obtain a duality result via the Borel transform and to prove existence and approximation results for convolution equations. The results we prove generalize previous results of this type due to B. Malgrange: Existence et approximation des équations aux dérivées partielles et des équations des convolutions. Annales...

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

Erhan Çalışkan (2007)

Czechoslovak Mathematical Journal

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.

Integral polynomials on Banach spaces not containing 1

Raffaella Cilia, Joaquín M. Gutiérrez (2010)

Czechoslovak Mathematical Journal

We give new characterizations of Banach spaces not containing 1 in terms of integral and p -dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.

Linearity in non-linear problems.

Richard Aron, Domingo García, Manuel Maestre (2001)

RACSAM

Estudiamos algunas situaciones donde encontramos un problema que, a primera vista, parece no tener solución. Pero, de hecho, existe un subespacio vectorial grande de soluciones del mismo.

Lower bounds for norms of products of polynomials on L p spaces

Daniel Carando, Damián Pinasco, Jorge Tomás Rodríguez (2013)

Studia Mathematica

For 1 < p < 2 we obtain sharp lower bounds for the uniform norm of products of homogeneous polynomials on L p ( μ ) , whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in infinite-dimensional settings). The result also holds for the Schatten classes p . For p > 2 we present some estimates on the constants involved.

M-ideals of homogeneous polynomials

Verónica Dimant (2011)

Studia Mathematica

We study the problem of whether w ( E ) , the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if w ( E ) is an M-ideal in (ⁿE), then w ( E ) coincides with w 0 ( E ) (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if w ( E ) = w 0 ( E ) and (E) is an M-ideal in...

Multilinear Hölder-type inequalities on Lorentz sequence spaces

Daniel Carando, Verónica Dimant, Pablo Sevilla-Peris (2009)

Studia Mathematica

We establish Hölder-type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.

Multilinear operators on C ( K , X ) spaces

Ignacio Villanueva (2004)

Czechoslovak Mathematical Journal

Given Banach spaces  X , Y and a compact Hausdorff space  K , we use polymeasures to give necessary conditions for a multilinear operator from C ( K , X ) into  Y to be completely continuous (resp.  unconditionally converging). We deduce necessary and sufficient conditions for  X to have the Schur property (resp.  to contain no copy of  c 0 ), and for  K to be scattered. This extends results concerning linear operators.

Norm-attaining polynomials and differentiability

Juan Ferrera (2002)

Studia Mathematica

We give a polynomial version of Shmul'yan's Test, characterizing the polynomials that strongly attain their norm as those at which the norm is Fréchet differentiable. We also characterize the Gateaux differentiability of the norm. Finally we study those properties for some classical Banach spaces.

On multilinear generalizations of the concept of nuclear operators

Dahmane Achour, Ahlem Alouani (2010)

Colloquium Mathematicae

This paper introduces the class of Cohen p-nuclear m-linear operators between Banach spaces. A characterization in terms of Pietsch's domination theorem is proved. The interpretation in terms of factorization gives a factorization theorem similar to Kwapień's factorization theorem for dominated linear operators. Connections with the theory of absolutely summing m-linear operators are established. As a consequence of our results, we show that every Cohen p-nuclear (1 < p ≤ ∞ ) m-linear mapping...

On multilinear mappings attaining their norms.

Maria Acosta (1998)

Studia Mathematica

We show, for any Banach spaces X and Y, the denseness of the set of bilinear forms on X × Y whose third Arens transpose attains its norm. We also prove the denseness of the set of norm attaining multilinear mappings in the class of multilinear mappings which are weakly continuous on bounded sets, under some additional assumptions on the Banach spaces, and give several examples of classical spaces satisfying these hypotheses.

On scalar-valued nonlinear absolutely summing mappings

Daniel Pellegrino (2004)

Annales Polonici Mathematici

We investigate cases ("coincidence situations") in which every scalar-valued continuous n-homogeneous polynomial (or every continuous n-linear mapping) is absolutely (p;q)-summing. We extend some well known coincidence situations and obtain several non-coincidence results, inspired by a linear technique due to Lindenstrauss and Pełczyński.

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