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Kevin Beanland

Publications and source records attributed to Kevin Beanland.

At least 19 recordsLinked to original sources
arXiv2026

Transfinite Schreier families and cardinal invariants

We study the dependence of the transfinite Schreier hierarchy on the choice of fundamental sequences for the countable limit ordinals. With each Schreier family we associate an interval endpoint function. We prove that the bounding number is equal to $ω_1$ exactly when the fundamental sequences may be chosen so that these endpoint functions form an unbounded family in the eventual domination order. Equivalently, the hierarchy then satisfies the tail-covering property isolated by Shiliaev, or every infinite compact interval family has infinite intersection with some member of the hierarchy. We also prove that the dominating number is equal to $ω_1$ exactly when the fundamental sequences may be chosen so that every compact family of finite subsets of $\{2,3,\ldots\}$ is contained in one Schreier family. The latter equivalence combines Fremlin's cofinality theorem for compact subsets of the rationals with an absorption construction for compact families.

math.FA↗
arXiv2025

Lower-Order Refinements of Greedy Approximation

For two countable ordinals $α$ and $β$, a basis of a Banach space $X$ is said to be $(α, β)$-quasi-greedy if it is 1) quasi-greedy, 2) $\mathcal{S}_α$-unconditional but not $\mathcal{S}_{α+1}$-unconditional, and 3) $\mathcal{S}_β$-democratic but not $\mathcal{S}_{β+1}$-democratic. If $α$ or $β$ is replaced with $\infty$, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a $(0,0)$-quasi-greedy basis, an $(α, \infty)$-quasi-greedy basis, and an $(\infty, α)$-quasi-greedy basis. In this paper, we construct $(α, β)$-quasi-greedy bases for $β\le α+1$ (except the already solved case $α= β= 0$).

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arXiv2023

On Schreier-type Sets, Partitions, and Compositions

A nonempty set $A\subset\mathbb{N}$ is $\ell$-strong Schreier if $\min A\geqslant \ell|A|-\ell+1$. We define a set of positive integers to be sparse if either the set has at most two numbers or the differences between consecutive numbers in increasing order are non-decreasing. This note establishes a connection between sparse Schreier-type sets and (restricted) partition numbers. One of our results states that if $\mathcal{G}_{n,\ell}$ consists of partitions of $n$ that contain no parts in $\{2, \ldots, \ell\}$, and \begin{equation*} \mathcal{A}_{n,\ell} \ :=\ \{A\subset \{1, \ldots, n\}\,:\, n\in A, A\mbox{ is sparse and }\ell\mbox{-strong Schreier}\}, \end{equation*} then $$|\mathcal{A}_{n,\ell}|\ =\ |\mathcal{G}_{n-1,\ell}|, \quad n, \ell\in \mathbb{N}.$$ The special case $\mathcal{G}_{n-1, 1}$ consists of all partitions of $n-1$. Besides partitions, integer compositions are also investigated.

math.CO↗
arXiv2023

Counting Unions of Schreier Sets

A subset of positive integers $F$ is a Schreier set if it is non-empty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most $k$ Schreier sets. Also, for each positive integer $n$, let $(k\mathcal{S})^n$ be the collection of all sets in $k\mathcal{S}$ with the maximum element equal to $n$. It is well-known that the sequence $(|(1\mathcal{S})^n|)_{n=1}^\infty$ is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence $(|(k\mathcal{S})^n|)_{n=1}^\infty$ satisfies a linear recurrence for every positive $k$.

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arXiv2023

The depth of Tsirelson's norm

Tsirelson's norm $\|\cdot \|_T$ on $c_{00}$ is defined as the supremum over a certain collection of iteratively defined, monotone increasing norms $\|\cdot \|_k$. For each positive integer $n$, the value $j(n)$ is the least integer $k$ such that for all $x \in \mathbb{R}^n$ (here $\mathbb{R}^n$ is considered as a subspace of $c_{00}$), $\|x\|_T = \|x\|_k$. In 1989 Casazza and Shura asked what is the order of magnitude of $j(n)$. It is known that $j(n) \in \mathcal{O}(\sqrt{n})$. We show that this bound is tight, that is, $j(n) \in Ω(\sqrt{n})$. Moreover, we compute the tight order of magnitude for some norms being modifications of the original Tsirelson's norm.

math.FA↗
arXiv2022

Schreier families and $\mathcal{F}$-(almost) greedy bases

Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. In this paper, we introduce and characterize $\mathcal{F}$-(almost) greedy bases. Given such a family $\mathcal{F}$, a basis $(e_n)_n$ for a Banach space $X$ is called $\mathcal{F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb{N}$, and $G_m(x)$, we have $$\|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.$$ Here $G_m(x)$ is a greedy sum of $x$ of order $m$, and $\mathbb{K}$ is the scalar field. From the definition, any $\mathcal{F}$-greedy basis is quasi-greedy and so, the notion of being $\mathcal{F}$-greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal{F}$-greedy bases as being $\mathcal{F}$-unconditional, $\mathcal{F}$-disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal{F}$-almost greedy bases. Furthermore, we provide several examples of bases that are nontrivially $\mathcal{F}$-greedy. For a countable ordinal $α$, we consider the case $\mathcal{F}=\mathcal{S}_α$, where $\mathcal{S}_α$ is the Schreier family of order $α$. We show that for each $α$, there is a basis that is $\mathcal{S}_α$-greedy but is not $\mathcal{S}_{α+1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $α< β$, $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_α\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_β\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$$

math.FA↗
arXiv2022

Shrinking Schauder Frames and their Associated Bases

For a Banach space $X$ with a shrinking Schauder frame $(x_i,f_i)$ we provide an explicit method for constructing a shrinking associated basis. In the case that the minimal associated basis is not shrinking, we prove that every shrinking associated basis of $(x_i,f_i)$ dominates an uncountable family of incomparable shrinking associated bases of $(x_i,f_i)$. By adapting a construction of Pełczy{ń}ski, we characterize spaces with shrinking Schauder frames as spaces having the $w^*$-bounded approximation property.

math.FA↗
arXiv2021

Generalized Schreier sets, linear recurrence relation, Turán graphs

We prove a linear recurrence relation for a large family of generalized Schreier sets, which generalizes the Fibonacci recurrence proved by Bird and higher order Fibonacci recurrence proved by the second author et al. Furthermore, we show a relationship between Schreier-type sets and Turán graphs.

math.CO↗
arXiv2021

Surjective isometries on Banach sequence spaces: a survey

In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries and characterizations of the group of surjective isometries for various Banach spaces.

math.FA↗
arXiv2020

Closed ideals of operators on the Tsirelson and Schreier spaces

Let $\mathscr{B}(X)$ denote the Banach algebra of bounded operators on $X$, where~$X$ is either Tsirelson's Banach space or the Schreier space of order $n$ for some $n\in\mathbb N$. We show that the lattice of closed ideals of~$\mathscr{B}(X)$ has a very rich structure; in particular $\mathscr{B}(X)$ contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection $P_N\in\mathscr{B}(X)$ corresponding to each non-empty subset $N$ of $\mathbb N$. A closed ideal of $\mathscr{B}(X)$ is spatial if it is generated by $P_N$ for some $N$. We can now state our main conclusions as follows: i) the family of spatial ideals lying strictly between the ideal of compact operators and $\mathscr{B}(X)$ is non-empty and has no minimal or maximal elements; ii) for each pair $\mathscr{I}\subsetneqq\mathscr{J}$ of spatial ideals, there is a family $\{Γ_L\colon L\in Δ\}$, where the index set $Δ$ has the cardinality of the continuum, such that $Γ_L$ is an uncountable chain of spatial ideals, $\bigcupΓ_L$ is a closed ideal that is not spatial, and $$ \mathscr{I}\subsetneqq\mathscr{L}\subsetneqq\mathscr{J}\qquad\text{and}\qquad \overline{\mathscr{L}+\mathscr{M}} = \mathscr{J}$$ whenever $L,M\inΔ$ are distinct and $\mathscr{L}\inΓ_L$, $\mathscr{M}\inΓ_M$.

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arXiv2019

Uniformly Factoring Weakly Compact operators and Parametrized Dualization

This paper deals with the problem of when, given a collection $\mathcal C$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space $Z$ with a Schauder basis so that every element in $\mathcal C$ factors through $Z$ (or through a subspace of $Z$). A sample result is the existence of a reflexive space $Z$ with a Schauder basis so that for each separable Banach space $X$, each weakly compact operator from $X$ to $L_1$ factors through $Z$. We also prove the following descriptive set theoretical result: Let $\mathcal L$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal B$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then the assignment $A\in \mathcal B\to A^*$ can be realized by a Borel map $\mathcal B\to \mathcal L$.

math.FA↗
arXiv2019

A $ξ$-weak Grothendieck compactness principle

For $0\leqslant ξ\leqslant ω_1$, we define the notion of $ξ$-weakly precompact and $ξ$-weakly compact sets in Banach spaces and prove that a set is $ξ$-weakly precompact if and only if its weak closure is $ξ$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\leqslant ξ\leqslant ω_1$, we prove that a Banach space $X$ has the $ξ$-Schur property if and only if every $ξ$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.

math.FA↗
arXiv2019

The Schreier Space Does Not Have the Uniform $λ$-property

The $λ$-property and the uniform $λ$-property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space possesses the $λ$-property and asked if it has the uniform $λ$-property. In this paper, we show that Schreier space does not have the uniform $λ$-property. Furthermore, we show that the dual of the Schreier space does not have the uniform $λ$-property.

math.FA↗
arXiv2019

On the geometry of higher order Schreier spaces

For each countable ordinal $α$ let $\mathcal{S}_α$ be the Schreier set of order $α$ and $X_{\mathcal{S}_α}$ be the corresponding Schreier space of order $α$. In this paper we prove several new properties of these spaces. 1) If $α$ is non-zero then $X_{\mathcal{S}_α}$ possesses the $λ$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $α$ is non-zero and $1<p<\infty$ then the $p$-convexification $X^{p}_{\mathcal{S}_α}$ possesses the uniform $λ$-property of R. Aron and R. Lohman. 3) For each countable ordinal $α$ the space $X^*_{\mathcal{S}_α}$ has the $λ$-property. 4) For $n\in \mathbb{N}$, if $U:X_{\mathcal{S}_n}\to X_{\mathcal{S}_n}$ is an onto linear isometry then $Ue_i = \pm e_i$ for each $i \in \mathbb{N}$. Consequently, these spaces are light in the sense of Megrelishvili. The fact that for non-zero $α$, $X_{\mathcal{S}_α}$ is $(V)$-polyhedral and has the $λ$-property implies that each $X_{\mathcal{S}_α}$ is an example of space solving a problem of J. Lindenstrauss from 1966. The first example of such a space was given by C. De Bernardi in 2017 using a renorming of $c_0$.

math.FA↗
arXiv2017

Genericity and Universality for Operator Ideals

A bounded linear operator $U$ between Banach spaces is universal for the complement of some operator ideal $\mathfrak{J}$ if it is a member of the complement and it factors through every element of the complement of $\mathfrak{J}$. In the first part of this paper, we produce new universal operators for the complements of several ideals and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use Descriptive Set Theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal $\mathfrak{J}$ generic if anytime an operator $A$ has the property that every operator in $\mathfrak{J}$ factors through a restriction of $A$, then every operator between separable Banach spaces factors through a restriction of $A$. We prove that many of classical operator ideals are generic (i.e. strictly singular, weakly compact, Banach-Saks) and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W.B. Johnson which states that there is no universal operator for the complement of the ideal of completely continuous operators.

math.FA↗
arXiv2016

On a generalization of Bourgain's tree index

For a Banach space $X$, a sequence of Banach spaces $(Y_n)$, and a Banach space $Z$ with an unconditional basis, D. Alspach and B. Sari introduced a generalization of a Bourgain tree called a $(\oplus_n Y_n)_Z$-tree in $X$. These authors also prove that any separable Banach space admitting a $(\oplus_n Y_n)_Z$-tree with order $ω_1$ admits a subspace isomorphic to $(\oplus_n Y_n)_Z$. In this paper we give two new proofs of this result.

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