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Register a new userIdeals of operators on $(oplus ell^infty(n))_{ell^1}$
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Authors
Denny H. Leung
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The unique maximal ideal in the Banach algebra $L(E)$, $E = (oplus ell^infty(n))_{ell^1}$, is identified. The proof relies on techniques developed by Laustsen, Loy and Read and a dichotomy result for operators mapping into $L^1$ due to Laustsen, Odell, Schlumprecht and Zs{a}k.
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In this paper we consider the following problem: Let $X_k$, be a Banach space with a normalized basis $(e_{(k,j)})_j$, whose biorthogonals are denoted by $(e_{(k,j)}^*)_j$, for $kinmathbb{N}$, let $Z=ell^infty(X_k:kinmathbb{N})$ be their $ell^infty$-sum, and let $T:Zto Z$ be a bounded linear operator, with a large diagonal, i.e. $$inf_{k,j} big|e^*_{(k,j)}(T(e_{(k,j)})big|>0.$$ Under which condition does the identity on $Z$ factor through $T$? The purpose of this paper is to formulate general conditions for which the answer is positive.
If alpha and beta are countable ordinals such that beta eq 0, denote by tilde{T}_{alpha,beta} the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_{0}}, 1/2 sup sum_{i=1}^{j}||E_{i}x||}, where the supremum is taken over all finite subsets E_{1},...,E_{j} of $mathbb{N}$ such that $E_{1}<...<E_{j}$ and {min E_{1},...,min E_{j}} in S_beta. It is shown that the Bourgain $ell^{1}$-index of tilde{T}_{alpha,beta} is omega^{alpha+beta.omega}. In particular, if alpha =omega^{alpha_{1}}. m_{1}+...+omega^{alpha_{n}}. m_{n} in Cantor normal form and alpha_{n} is not a limit ordinal, then there exists a Banach space whose ell^{1}-index is omega^{alpha}.
Suppose that (F_n)_{n=0}^{infty} is a sequence of regular families of finite subsets of N such that F_0 contains all singletons, and (theta _n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). In this paper, we compute the Bourgain ell^1 - index of the mixed Tsirelson space T(F_0,(theta_n, F_n)_{n=1}^{infty}). As a consequence, it is shown that if eta is a countable ordinal not of the form omega^xi for some limit ordinal xi, then there is a Banach space whose ell^1-index is omega^eta . This answers a question of Judd and Odell.
We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted $ell^1$ space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted $ell^1$ space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions, and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.
Suppose that (F_n)_{n=1}^{infty} is a sequence of regular families of finite subsets of N and (theta_n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T[(theta_{n}, F_n)_{n=1}^{infty}] is the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_0}, sup_n sup theta_n sum_{i=1}^{j}||E_{i}x||}, where the last supremum is taken over all finite subsets E_{1},...,E_{j} of N such that E_1 < >... <E_j and {min E_1,...,min E_j} in F_n. Necessary and sufficient conditions are obtained for the existence of higher order ell ^1-spreading models in every subspace generated by a subsequence of the unit vector basis of T[(theta_{n}, F_n)_{n=1}^{infty}.
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