Analytic P-ideals and Banach spaces

Borodulin–Nadzieja, Piotr; Farkas, Barnabás

doi:10.1016/j.jfa.2020.108702

Cited by 13 publications

(13 citation statements)

References 42 publications

(43 reference statements)

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“…Section 3. The results in [5] suggest that the study of the interplay between the theory of analytic P-ideals and their representability may have some relevant yet unexploited potential for the study of the geometry of Banach spaces.…”

Section: Resultsmentioning

confidence: 97%

Most numbers are not normal

Aveni¹,

Leonetti²

2021

Preprint

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We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers x ∈ (0, 1] with the following property is comeager: for all integers b ≥ 2 and k ≥ 1, the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen in [Math. Proc. Cambridge Philos. Soc. 137 (2004), 43-53]. We provide analogues in the context of analytic P-ideals and regular matrices.

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Section: Resultsmentioning

confidence: 97%

Most numbers are not normal

Aveni¹,

Leonetti²

2021

Preprint

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“…Remark 2.19. In some cases, it is easy to provide examples of matrices which satisfy (6). Indeed, suppose that T ∈ B(X, Y ) and I ⊆ J .…”

Section: In Addition If Eachmentioning

confidence: 99%

“…An additional motivation comes from the fact that that the study of ideals on countable sets and their representability may have some relevant potential for the study of the geometry of Banach spaces, see e.g. [6,7,24,27].…”

Section: Introductionmentioning

confidence: 99%

Regular matrices of unbounded linear operators

Leonetti¹

2022

Preprint

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Let X, Y be Banach spaces, and fix a linear operator T ∈ L(X, Y ), and ideals I, J on ω. We obtain Silverman-Toeplitz type theorems on matrices A = (A n,k : n, k ∈ ω) of linear operators in L(X, Y ), so thatfor every X-valued sequence x = (x 0 , x 1 , . . .) which is I-convergent [and bounded]. This allows us to establish the relationship between the classical Silverman-Toeplitz characterization of regular matrices and its multidimensional analogue for double sequences, its variant for matrices of linear operators, and the recent version (for the scalar case) in the context of ideal convergence. As byproducts, we obtain characterizations of several matrix classes and a generalization of the classical Hahn-Schur theorem. In the proofs we will use an ideal version of the Banach-Steinhaus theorem which has been recently obtained by De Bondt and Vernaeve in [J. Math. Anal. Appl. 495 (2021)].

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“…Moreover, for instance, it is known that an ideal is representable in if and only if it is an intersection of countably many summable ideals [5, Example 3.8] (i.e., ideals of the form for some such that ); for more on this notion see [5]. It is worth mentioning that Borodulin-Nadzieja and Farkas, using representability of ideals in Banach spaces, constructed a new example of a Banach space [4, Example 5.9], and strengthened Mazur’s Lemma [4, Corollary 7.6], which is a basic tool in Banach space theory (they were able to specify the form of the convex combination in Mazur’s Lemma). This suggests that studying the interplay between representability and theory of analytic P-ideals may have some relevant yet unexploited potential for the study of the geometry of Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

Density-Like and Generalized Density Ideals

Kwela¹,

Leonetti²

2021

J. symb. log.

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We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80 (2015), pp. 1268–1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

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Analytic P-ideals and Banach spaces

Cited by 13 publications

References 42 publications

Most numbers are not normal

Most numbers are not normal

Regular matrices of unbounded linear operators

Density-Like and Generalized Density Ideals

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