arXiv subjects

Philip A. H. Brooker

Publications and source records attributed to Philip A. H. Brooker.

6 recordsLinked to original sources

arXiv2019

Absolutely convex sets of large Szlenk index

Let $X$ be a Banach space and $K$ an absolutely convex, weak$^\ast$-compact subset of $X^\ast$. 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 $Y$ such that $Y$ has a basis and the Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is separable, then $X$ also admits subspace $Z$ such that the quotient $X/Z$ has a basis and the Szlenk indices of $X/Z$ are comparable to the Szlenk indices of $X$. 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 $\ell_1$ to $\ell_\infty$ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.

math.FA↗

arXiv2017

Non-Asplund Banach spaces and operators

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.

math.FA↗

arXiv2012

Szlenk and $w^\ast$-dentability indices of the Banach spaces $C([0,α])$

Let $α$ be an infinite ordinal and $γ$ the unique ordinal satisfying $ω^{ω^γ}\leq α< ω^{ω^{γ+1}}$. We show that the Banach space $C([0,\,α])$ of all continuous scalar-valued functions on the compact ordinal interval $[0,\,α]$ has Szlenk index equal to $ω^{γ+1}$ and $w^\ast$-dentability index equal to $ω^{1+γ+1}$.

math.FA↗

arXiv2011

Asplund operators and the Szlenk index

For $α$ an ordinal, we investigate the class $\mathscr{SZ}_α$ consisting of all operators whose Szlenk index is an ordinal not exceeding $ω^α$. We show that each class $\mathscr{SZ}_α$ is a closed operator ideal and study various operator ideal properties for these classes. The relationship between the classes $\mathscr{SZ}_α$ and several well-known closed operator ideals is investigated and quantitative factorization results in terms of the Szlenk index are obtained for the class of Asplund operators.

math.FA↗

arXiv2010

Operator ideals associated with the Szlenk index

For $α$ an ordinal, we investigate the class $\mathscr{SZ}_α$ consisting of all operators whose Szlenk index is an ordinal not exceeding $ω^α$. Our main result is that $\mathscr{SZ}_α$ is a closed, injective, surjective operator ideal for each $α$. We also study the relationship between the classes $\mathscr{SZ}_α$ and several well-known closed operator ideals.

math.FA↗

arXiv2010

Direct sums and the Szlenk index

For $α$ an ordinal and $1<p<\infty$, we determine a necessary and sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk index not exceeding $ω^α$. It follows from our results that the Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural way by the behaviour of the $ε$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums.

math.FA↗