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| 2010-01-29T03:22:30Z | | Subspace hyperciclicity | | A bounded linear operator T on Hilbert space is subspace-hypercyclic for a
subspace M if there exists a vector whose orbit under T intersects the subspace
in a relatively dense set. We construct examples to show that
subspace-hypercyclicity is interesting, including a nontrivial
subspace-hypercyclic operator that is not hypercyclic. There is a Kitai-like
criterion that implies subspace-hypercyclicity and although the spectrum of a
subspace-hypercyclic operator must intersect the unit circle, not every
component of the spectrum will do so. We show that, like hypercyclicity,
subspace-hypercyclicity is a strictly infinite-dimensional phenomenon.
Additionally, compact or hyponormal operators can never be
subspace-hypercyclic.
| | Blair Madore Rubén A. Martínez Avendaño |
| 2009-07-30T12:23:25Z | | Multiple pattern classification by sparse subspace decomposition | | A robust classification method is developed on the basis of sparse subspace
decomposition. This method tries to decompose a mixture of subspaces of
unlabeled data (queries) into class subspaces as few as possible. Each query is
classified into the class whose subspace significantly contributes to the
decomposed subspace. Multiple queries from different classes can be
simultaneously classified into their respective classes. A practical greedy
algorithm of the sparse subspace decomposition is designed for the
classification. The present method achieves high recognition rate and robust
performance exploiting joint sparsity.
| | Tomoya Sakai |
| 2000-02-28T19:10:43Z | | Complemented subspaces of products of Banach spaces | | We show that complemented subspaces of uncountable products of Banach spaces
are products of complemented subspaces of countable subproducts.
| | Alex Chigogidze |
| 1990-07-20T18:06:00Z | | On the complemented subspaces of X_p | | In this paper we prove some results related to the problem of isomorphically
classifying the complemented subspaces of $X_{p}$. We characterize the
complemented subspaces of $X_{p}$ which are isomorphic to $X_{p}$ by showing
that such a space must contain a canonical complemented subspace isomorphic to
$X_{p}.$ We also give some characterizations of complemented subspaces of
$X_{p}$ isomorphic to $\ell_{p}\oplus \ell_{2}.$
| | Dale E. Alspach |
| 2010-06-21T12:12:27Z | | Online Identification and Tracking of Subspaces from Highly Incomplete
Information | | This work presents GROUSE (Grassmanian Rank-One Update Subspace Estimation),
an efficient online algorithm for tracking subspaces from highly incomplete
observations. GROUSE requires only basic linear algebraic manipulations at each
iteration, and each subspace update can be performed in linear time in the
dimension of the subspace. The algorithm is derived by analyzing incremental
gradient descent on the Grassmannian manifold of subspaces. With a slight
modification, GROUSE can also be used as an online incremental algorithm for
the matrix completion problem of imputing missing entries of a low-rank matrix.
GROUSE performs exceptionally well in practice both in tracking subspaces and
as an online algorithm for matrix completion.
| | Laura Balzano Robert Nowak Benjamin Recht |
| 1999-02-01T16:12:14Z | | Operators between subspaces and quotients of L1 | | We provide an unified approach of results of L. Dor on the complementation of
the range, and of D. Alspach on the nearness from isometries, of small into
isomorphisms of L1. We introduce the notion of small subspace of L1 and show
lifting theorems for operators between quotients of L1 by small subspaces. We
construct a subspace of L1 which shows that extension of isometries from
subspaces of L1 to the whole space are no longer true for isomorphisms, and
that nearly isometric isomorphisms from subspaces of L1 into L1 need not be
near from any isometry.
| | G. Godefroy N. Kalton D. Li |
| 2003-04-02T12:33:23Z | | Ergodic Banach Spaces | | We show that any Banach space contains a continuum of non isomorphic
subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a
space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of
$X$, and show that every Banach space is either ergodic or contains a subspace
with an unconditional basis $ which is complementably universal for the family
of its block-subspaces. We also use our methods to get uniformity results; for
example, in combination with a result of B. Maurey, V. Milman and N.
Tomczak-Jaegermann, we show that an unconditional basis of a Banach space, of
which every block-subspace is complemented, must be asymptotically $c_0$ or
$l_p$.
| | Valentin Ferenczi Christian Rosendal |
| 2008-07-14T16:55:14Z | | $n$-Subspaces in linear and unitary spaces | | We study a relation between brick $n$-tuples of subspaces of a finite
dimensional linear space, and irreducible $n$-tuples of subspaces of a finite
dimensional Hilbert (unitary) space such that a linear combination, with
positive coefficients, of orthogonal projections onto these subspaces equals
the identity operator. We prove that brick systems of one-dimensional subspaces
and the systems obtained from them by applying the Coxeter functors (in
particular, all brick triples and quadruples of subspaces) can be unitarized.
For each brick triple and quadruple of subspaces, we describe sets of
characters that admit a unitarization.
| | Yu. S. Samoilenko D. Y. Yakymenko |
| 2006-06-19T12:54:10Z | | Inner Ideals and Intrinsic Subspaces of Linear Pair Geometries | | We introduce the notion of intrinsic subspaces of linear and affine pair
geometries, which generalizes the one of projective subspaces of projective
spaces. We prove that, when the affine pair geometry is the projective geometry
of a Lie algebra introduced in [Bertram-Neeb, J. Alg. 277], such intrinsic
subspaces correspond to inner ideals in the associated Jordan pair, and we
investigate the case of intrinsic subspaces defined by the Peirce-decomposition
which is related to 5-gradings of the projective Lie algebra. These examples,
as well as the examples of general and Lagrangian flag geometries, lead to the
conjecture that geometries of intrinsic subspaces tend to be themselves linear
pair geometries.
| | Wolfgang Bertram Harald Loewe |
| 2007-11-08T20:48:00Z | | Banach spaces without minimal subspaces | | We prove three new dichotomies for Banach spaces \`a la W. T. Gowers'
dichotomies. The three dichotomies characterise respectively the spaces having
no minimal subspaces, having no subsequentially minimal basic sequences, and
having no subspaces crudely finitely representable in all of their subspaces.
We subsequently use these results to make progress on the program of Gowers of
classifying Banach spaces by finding characteristic spaces present in every
space. Also, the results are used to embed any partial order of size $\aleph_1$
into the subspaces of any space without a minimal subspace ordered by
isomorphic embeddability. Finally, we analyse several examples of spaces and
classify them according to which side of the dichotomies they fall.
| | Valentin Ferenczi Christian Rosendal |
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