Nonexistence of certain universal polynomials between Banach spaces

Cilia, Raffaella; Gutiérrez, Joaquín M.

doi:10.1016/j.jmaa.2015.02.072

Cited by 3 publications

(9 citation statements)

References 20 publications

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“…Other universality results besides those mentioned above can be found in the work of Brooker [3], Cilia and Gutiérrez [5], Dilworth [9], Girardi and Johnson [14], Hinrichs and Pietsch [17], Oikhberg [25], and the Handbook survey on operator ideals by Diestel, Jarchow and Pietsch [6]. Recent work of R. Causey on the existence of non-Asplund spaces of type 2, in a similar vein to work of Pisier and Xu [28], has produced another construction of a universal non-Asplund operator using interpolation techniques [4].…”

Section: Introductionmentioning

confidence: 69%

Non-Asplund Banach spaces and operators

Brooker¹

2017

Journal of Functional Analysis

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Let $W$ and $Z$ be Banach spaces such that $Z$ is separable and let $R:W\longrightarrow Z$ be a (continuous, linear) operator. We study consequences of the adjoint operator $R^\ast$ having non-separable range. From our main technical result we obtain applications to the theory of basic sequences and the existence of universal operators for various classes of operators between Banach spaces. We also obtain an operator-theoretic characterisation of separable Banach spaces with non-separable dual.Comment: Accepted for publication by Journal of Functional Analysis. Version 3 features a number of minor corrections as well as a change to the title of the paper (which was originally 'Banach spaces and operators with non-separable dual'). arXiv admin note: text overlap with arXiv:1612.0812

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Section: Introductionmentioning

confidence: 69%

Non-Asplund Banach spaces and operators

Brooker¹

2017

Journal of Functional Analysis

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“…The following result can be found in [, Proposition 2.4]. See also the proof of [, Proposition 1] which works for

P_{U}

with an immediate argument.…”

Section: Introductionmentioning

confidence: 83%

“…The problem of the existence of universal polynomials seems to be very different with respect to the linear case and the lack of linearity introduces a degree of difficulty. In we have proved that there are neither a universal non‐compact polynomial nor a universal non‐unconditionally converging polynomial. In the present paper we investigate the existence of a universal non‐weakly compact polynomial between Banach spaces.…”

Section: Introductionmentioning

confidence: 90%

“…Moreover, we show the nonexistence of universal non‐Rosenthal, non‐Asplund, non‐limited, non‐Grothendieck, and non‐Dunford–Pettis polynomials. The techniques used here are in most cases different from those of .…”

Section: Introductionmentioning

confidence: 99%

“…We recall some definitions and results from . Definition Let

scriptU

be a closed surjective operator ideal.…”

Section: Introductionmentioning

confidence: 99%

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Universal mappings for certain classes of operators and polynomials between Banach spaces

Cilia

Gutiérrez

2017

Mathematische Nachrichten

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A well-known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non-weakly compact operator between Banach spaces. We show the existence of universal non-Rosenthal, non-limited, and non-Grothendieck operators.We also prove that there does not exist a universal non-Dunford-Pettis operator, but there is a universal class of non-Dunford-Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non-weakly compact polynomials, there does not exist a universal polynomial. K E Y W O R D SIdeals of homogeneous polynomials, surjective operator ideals, universal operator, universal polynomial M S C ( 2 0 1 0 ) Johnson [27] showed in 1971 that the formal identity operator 1 → ∞ is universal for the class of non-compact operators. These results are useful in order to prove that a given operator is non-weakly compact (respectively, non-compact). In 1997, M. Girardi and W. B. Johnson [21] proved that there does not exist a universal non-completely continuous operator, but there is a class  of universal non-completely continuous operators, that is, for every non-completely continuous operator , there is some member of  that factors through .Here we prove the existence of universal operators for the classes of non-Rosenthal, non-Grothendieck, and non-limited operators and the existence of a universal class of non-Dunford-Pettis operators (see below the definitions of all such classes

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Absolutely convex sets of large Szlenk index

Brooker

2016

Preprint

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Let X be a Banach space and K an absolutely convex, weak * -compact subset of X. We study consequences of K having a large or undefined Szlenk index, and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if X has a countable Szlenk index then X admits a subspace with a basis and with Szlenk indices comparable to the Szlenk indices of X. If X is separable, then X also admits a quotient with these same properties. We also show that for a given ordinal ξ the class of operators whose Szlenk index is not an ordinal less than or equal to ξ admits a universal element if and only if ξ < ω 1 ; W.B. Johnson's theorem that the formal identity map from ℓ 1 to ℓ ∞ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.

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Nonexistence of certain universal polynomials between Banach spaces

Cited by 3 publications

References 20 publications

Non-Asplund Banach spaces and operators

Non-Asplund Banach spaces and operators

Universal mappings for certain classes of operators and polynomials between Banach spaces

Absolutely convex sets of large Szlenk index

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