130 research outputs found
On Uniformly finitely extensible Banach spaces
We continue the study of Uniformly Finitely Extensible Banach spaces (in
short, UFO) initiated in Moreno-Plichko, \emph{On automorphic Banach spaces},
Israel J. Math. 169 (2009) 29--45 and Castillo-Plichko, \emph{Banach spaces in
various positions.} J. Funct. Anal. 259 (2010) 2098-2138. We show that they
have the Uniform Approximation Property of Pe\l czy\'nski and Rosenthal and are
compactly extensible. We will also consider their connection with the
automorphic space problem of Lindenstrauss and Rosenthal --do there exist
automorphic spaces other than and ?-- showing that a space
all whose subspaces are UFO must be automorphic when it is Hereditarily
Indecomposable (HI), and a Hilbert space when it is either locally minimal or
isomorphic to its square. We will finally show that most HI --among them, the
super-reflexive HI space constructed by Ferenczi-- and asymptotically
spaces in the literature cannot be automorphic.Comment: This paper is to appear in the Journal of Mathematical Analysis and
Application
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
The affine rank minimization problem consists of finding a matrix of minimum
rank that satisfies a given system of linear equality constraints. Such
problems have appeared in the literature of a diverse set of fields including
system identification and control, Euclidean embedding, and collaborative
filtering. Although specific instances can often be solved with specialized
algorithms, the general affine rank minimization problem is NP-hard. In this
paper, we show that if a certain restricted isometry property holds for the
linear transformation defining the constraints, the minimum rank solution can
be recovered by solving a convex optimization problem, namely the minimization
of the nuclear norm over the given affine space. We present several random
ensembles of equations where the restricted isometry property holds with
overwhelming probability. The techniques used in our analysis have strong
parallels in the compressed sensing framework. We discuss how affine rank
minimization generalizes this pre-existing concept and outline a dictionary
relating concepts from cardinality minimization to those of rank minimization
Banach spaces in various positions
AbstractWe formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ℓp,Lp,L1,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c0 or ℓ2? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c0 or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L∞ or a superreflexive type 2 Banach lattice
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