- Banach - mathdept.okstate.edu

Abstract of a paper by Radoslaw Adamczak, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Sharp bounds on the rate of convergence of the empirical covariance matrix" by Radoslaw Adamczak, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann. Abstract: Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in S^{n-1}} \Big|\frac{1/N}\sum_{i=1}^N (|<X_i, x>|^2 - \E|<X_i, x>|^2\r)\Big| \leq C \sqrt{\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $(<X_i,x>)_{i\leq N, x\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\sqrt n)_{i\leq N}$ exhibit uniformly a sub-exponential decay. As a consequence, if $A$ denotes the random matrix with columns $(X_i)$, then with overwhelming probability, the extremal singular values $\lambda_{\rm min}$ and $\lambda_{\rm max}$ of $AA^\top$ satisfy the inequalities $ 1 - C\sqrt{{n/N}} \le {\lambda_{\rm min}/N} \le \frac{\lambda_{\rm max}/N} \le 1 + C\sqrt{{n/N}} $ which is a quantitative version of Bai-Yin theorem \cite{BY} known for random matrices with i.i.d. entries. Archive classification: math.PR math.FA Mathematics Subject Classification: 52A20, 46B09, 52A21 (Primary) 15A52, 60E15 (Secondary) Submitted from: radamcz(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1012.0294 or http://arXiv.org/abs/1012.0294

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Abstract of a paper by Rafal Latala
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Order statistics and concentration of l_r norms for log-concave vectors" by Rafal Latala. Abstract: We establish upper bounds for tails of order statistics of isotropic log-concave vectors and apply them to derive a concentration of l_r norms of such vectors. Archive classification: math.PR math.FA Mathematics Subject Classification: 60E15 (52A38, 60B11) Remarks: 17 pages Submitted from: rlatala(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.6610 or http://arXiv.org/abs/1011.6610

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Abstract of a paper by O.F.K. Kalenda, H. Pfitzner and J. Spurny
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "On quantification of weak sequential completeness" by O.F.K. Kalenda, H. Pfitzner and J. Spurny. Abstract: We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in $L$-embedded Banach spaces, improving in particular an inequality of G.~Godefroy, N.~Kalton and D.~Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space $X$ with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G.~Godefroy. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 9 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.6553 or http://arXiv.org/abs/1011.6553

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Abstract of a paper by Gustavo Corach and Alejandra Maestripieri
by alspach＠math.okstate.edu 21 Dec '10

21 Dec '10

This is an announcement for the paper "Products of orthogonal projections and polar decompositions" by Gustavo Corach and Alejandra Maestripieri. Abstract: We characterize the sets $\XX$ of all products $PQ$, and $\YY$ of all products $PQP$, where $P,Q$ run over all orthogonal projections and we solve the problems \newline $\arg\min\{\|P-Q\|: (P,Q) \in \cal Z\}$, for $\cal Z=\XX$ or $\YY.$ We also determine the polar decompositions and Moore-Penrose pseudoinverses of elements of $\XX.$ Archive classification: math.FA Mathematics Subject Classification: 47A05 Submitted from: gcorach(a)fi.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.5237 or http://arXiv.org/abs/1011.5237

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Announcement of Meeting
by Dale Alspach 10 Dec '10

10 Dec '10

Announcement of Meeting: SET THEORETIC TECHNIQUES IN FUNCTIONAL ANALYSIS To be held in Castro Urdiales (Cantabria) Spain, from February 21 to February 24, 2011. Organized by Jesús M. F. Castillo (Univ. of Extremadura) and Manuel González (Univ. of Cantabria), in collaboration with the CIEM (International Center for Matematical Encounters). The meeting includes six mini-courses lectured by: Antonio Avilés (Univ. Murcia, Spain) (2 hours) Push-out constructions in Banach spaces and Boolean algebras Valentin Ferenczi (Univ. Sao Paulo, Brasil) (2 hours) Groups of isometries on Banach spaces Piotr Koszmider (Univ. Lodz, Poland) (2 hours) Some applications of set-theoretic topological methods in C(K) spaces Wieslaw Kubis (Univ. Praga, Czech Republic) (2 hours) Category-theoretic methods for constructing universal Banach spaces Jordi López Abad (ICMAT-CSIC, Madrid, Spain) (2 hours) Banach Spaces and Ramsey Theory: some open problems Stevo Todorcevic (Univ. Toronto, Canada) (3 hours) Combinatorial dichotomies in set theory and their applications to analysis Participants will have the possibility of delivering short lectures of 20 or 30 minutes. Registration for the meeting can be done through the web-site: http://www.ciem.unican.es/encuentros/banach2011. There is no registration fee. Additional information can be found in that web-site. The meeting is supported by CIEM, Universidad de Cantabria, Ayuntamiento de Castro Urdiales and Ingenio Mathematica. Antonio MArtínez-Abejón (University of Oviedo, Spain)

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Abstract of a paper by Miguel Martin, Javier Meri and Mikhail Popov
by alspach＠math.okstate.edu 23 Nov '10

23 Nov '10

This is an announcement for the paper "On the numerical radius of operators in Lebesgue spaces" by Miguel Martin, Javier Meri and Mikhail Popov. Abstract: We show that the absolute numerical index of the space $L_p(\mu)$ is $p^{-\frac{1}{p}} q^{-\frac{1}{q}}$ (where $1/p+1/q=1$). In other words, we prove that $$ \sup\left\{\int |x|^{p-1}|Tx|\, d\mu \, : \ x\in L_p(\mu),\,\|x\|_p=1\right\} \,\geq \,p^{-\frac{1}{p}} q^{-\frac{1}{q}}\,\|T\| $$ for every $T\in \mathcal{L}(L_p(\mu))$ and that this inequality is the best possible when the dimension of $L_p(\mu)$ is greater than one. We also give lower bounds for the best constant of equivalence between the numerical radius and the operator norm in $L_p(\mu)$ for atomless $\mu$ when restricting to rank-one operators or narrow operators. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 47A12 Remarks: 14 pages Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.4785 or http://arXiv.org/abs/1011.4785

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Abstract of a paper by Matthew Tarbard
by alspach＠math.okstate.edu 23 Nov '10

23 Nov '10

This is an announcement for the paper "Hereditarily indecomposable, separable L_\infty spaces with \ell_1 dual having few operators, but not very few operators" by Matthew Tarbard. Abstract: Given a natural number $k \geq 2$, we construct a hereditarily indecomposable, $\mathscr{L}_{\infty}$ space, $X_k$ with dual isomorphic to $\ell_1$. We exhibit a non-compact, strictly singular operator $S$ on $X_k$, with the property that $S^k = 0$ and $S^j (0 \leq j \leq k-1)$ is not a compact perturbation of any linear combination of $S^l, l \neq j$. Moreover, every bounded linear operator on this space has the form $\sum_{i=0}^{k-1} \lambda_i S^i +K$ where the $\lambda_i$ are scalars and $K$ is compact. In particular, this construction answers a question of Argyros and Haydon ( "A hereditarily indecomposable space that solves the scalar-plus-compact problem"). Archive classification: math.FA Submitted from: matthew.tarbard(a)sjc.ox.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.4776 or http://arXiv.org/abs/1011.4776

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Abstract of a paper by D. Azagra, R. Fry and L. Keener
by alspach＠math.okstate.edu 23 Nov '10

23 Nov '10

This is an announcement for the paper "Approximation of functions and their derivatives by analytic maps on certain Banach spaces" by D. Azagra, R. Fry and L. Keener. Abstract: Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let f:X→R be bounded, Lipschitz, and C¹ with uniformly continuous derivative. Then for each {\epsilon}>0, there exists an analytic function g:X→R with |g-f|<{\epsilon} and ‖g′-f′‖<{\epsilon}. Archive classification: math.FA Remarks: 17 pages Submitted from: rfry(a)tru.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.4613 or http://arXiv.org/abs/1011.4613

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Abstract of a paper by Dale E. Alspach and Eloi Medina Galego
by alspach＠math.okstate.edu 23 Nov '10

23 Nov '10

This is an announcement for the paper "Geometry of the Banach spaces C(beta mathbb N times K, l_p) for compact metric spaces K" by Dale E. Alspach and Eloi Medina Galego . Abstract: In this paper we provide the complete isomorphic classification of the spaces C(beta mathbb N times K, l_p) of all continuous l_p-valued functions, 1 <= p < infinity, defined on the topological product of the Stone-Cech compactification of the natural numbers mathbb N and an arbitrary infinite compact metric space K. In order to do this, we first prove that c_0 is the only infinite dimensional separable C(K) space, Z, up to an isomorphism, which satisfies each one of the following statements: (1) Z is a quotient of C(beta mathbb N, l_p) for every 1< p< infinity. (2) Z is isomorphic to a complemented subspace of C(beta mathbb N, l_1). (3) C(beta mathbb N, l_p) is isomorphic to the injective tensor product of itself and Z, for every 1 <= p < infinity. Archive classification: math.FA Mathematics Subject Classification: 46B Remarks: 17 pages Submitted from: alspach(a)math.okstate.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.3261 or http://arXiv.org/abs/1011.3261

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Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 23 Nov '10

23 Nov '10

This is an announcement for the paper "Some remarks on stronger versions of the Boundary Problem for Banach spaces" by Jan-David Hardtke. Abstract: Let $X$ be a real Banach space. A subset $B$ of the dual unit sphere of $X$ is said to be a boundary for $X$, if every element of $X$ attains its norm on some functional in $B$. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of $X$ which is compact in the topology of pointwise convergence on $B$ is already weakly compact. This problem was recently solved by H.Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others. Archive classification: math.FA Mathematics Subject Classification: 46A50, 46B50 Remarks: 14 pages Submitted from: hardtke(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1011.2372 or http://arXiv.org/abs/1011.2372

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