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We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelsons well-known space to larger index sets. We prove that for every cardinal $kappa$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $kappa$ without subsymmetric basic sequences. As for Tsirelsons space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.
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We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $aleph_omega$ contains an unconditional basic sequence.
We study the different horospherical Radon transforms that arise by regarding a homogeneous tree T as a simplicial complex whose simplices are vertices V, edges E or flags F (flags are oriented edges). The ends (infinite geodesic rays starting at a reference vertex) provide a boundary $Omega$ for the tree. Then the horospheres form a trivial principal fiber bundle with base $Omega$ and fiber $mathZ$. There are three such fiber bundles, consisting of horospheres of vertices, edges or flags, but they are isomorphic: however, no isomorphism between these fiber bundles maps special sections to special sections (a special section consists of the set of horospheres through a given vertex, edge or flag). The groups of automorphisms of the fiber bundles contain a subgroup $A$ of parallel shifts, analogous to the Cartan subgroup of a semisimple group. The normalized eigenfunctions of the Laplace operator on T are boundary integrals of complex powers of the Poisson kernel, that is characters of $A$, and are matrix coefficients of representations induced from $A$ in the sense of Mackey, the so-called spherical representations. The vertex-horospherical Radon transform consists of summation over V in each vertex-horosphere, and similarly for edges or flags. We prove inversion formulas for all these Radon transforms, and give applications to harmonic analysis and the Plancherel measure on T. We show via integral geometry that the spherical representations for vertices and edges are equivalent. Also, we define the Radon back-projections and find the inversion operator of each Radon transform by composing it with its back-projection. This gives rise to a convolution operator on T, whose symbol is obtained via the spherical Fourier transform, and its reciprocal is the symbol of the Radon inversion formula.
Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p in [1,infty]$) $Z(ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(ell_infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X setminus Z(X)$ and its algebraic genericity.
We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $om_n$ without unconditional subsequences. This together with a result of cite{Do-Lo-To} shows that $om_omega$ is the minimal cardinal $kappa$ that could possibly have the property that every weakly null $kappa$-sequence has an infinite unconditional basic subsequence . We also prove that for every cardinal number $ka$ which is smaller than the first $om$-Erdos cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either $c_0$ or $ell_p$, with $pge 1$.
We show that every subsymmetric Schauder basis $(e_j)$ of a Banach space $X$ has the factorization property, i.e. $I_X$ factors through every bounded operator $Tcolon Xto X$ with a $delta$-large diagonal (that is $inf_j |langle Te_j, e_j^*rangle| geq delta > 0$, where the $(e_j^*)$ are the biorthogonal functionals to $(e_j)$). Even if $X$ is a non-separable dual space with a subsymmetric weak$^*$ Schauder basis $(e_j)$, we prove that if $(e_j)$ is non-$ell^1$-splicing (there is no disjointly supported $ell^1$-sequence in $X$), then $(e_j)$ has the factorization property. The same is true for $ell^p$-direct sums of such Banach spaces for all $1leq pleq infty$. Moreover, we find a condition for an unconditional basis $(e_j)_{j=1}^n$ of a Banach space $X_n$ in terms of the quantities $|e_1+ldots+e_n|$ and $|e_1^*+ldots+e_n^*|$ under which an operator $Tcolon X_nto X_n$ with $delta$-large diagonal can be inverted when restricted to $X_sigma = [e_j : jinsigma]$ for a large set $sigmasubset {1,ldots,n}$ (restricted invertibility of $T$; see Bourgain and Tzafriri [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989). We then apply this result to subsymmetric bases to obtain that operators $T$ with a $delta$-large diagonal defined on any space $X_n$ with a subsymmetric basis $(e_j)$ can be inverted on $X_sigma$ for some $sigma$ with $|sigma|geq c n^{1/4}$.
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