- Banach - mathdept.okstate.edu

Abstract of a paper by Santeri Miihkinen
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "Strict singularity of a Volterra-type integral operator on $H^p$" by Santeri Miihkinen. Abstract: We prove that a Volterra-type integral operator $T_gf(z) = \int_0^z f(\zeta)g'(\zeta)d\zeta, \, z \in \mathbb D,$ defined on Hardy spaces $H^p, \, 1 \le p < \infty,$ fixes an isomorphic copy of $\ell^p,$ if the operator $T_g$ is not compact. In particular, this shows that the strict singularity of the operator $T_g$ coincides with the compactness of the operator $T_g$ on spaces $H^p.$ As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator $T_g$ on $H^1$. Archive classification: math.FA Mathematics Subject Classification: 47G10 (Primary) 30H10 (Secondary ) Remarks: 14 pages, 1 figure Submitted from: santeri.miihkinen(a)helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.08356 or http://arXiv.org/abs/1509.08356

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Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "On certain Opial-type results in Ces\`aro spaces of vector-valued functions" by Jan-David Hardtke. Abstract: Given a Banach space $X$, we consider Ces\`aro spaces $\text{Ces}_p(X)$ of $X$-valued functions over the interval $[0,1]$, where $1\leq p<\infty$. We prove that if $X$ has the Opial/uniform Opial property, then certain analogous properties also hold for $\text{Ces}_p(X)$. We also prove a result on the Opial/uniform Opial property of Ces\`aro spaces of vector-valued sequences. Archive classification: math.FA Mathematics Subject Classification: 46E40 46E30 46B20 Remarks: 15 pages, partial text overlap with arXiv:1403.2647 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/1509.08097 or http://arXiv.org/abs/1509.08097

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Abstract of a paper by Niushan Gao, Vladimir G. Troitsky, and Foivos Xanthos
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "Uo-convergence and its applications to Ces\`aro means in Banach lattices" by Niushan Gao, Vladimir G. Troitsky, and Foivos Xanthos. Abstract: A net $(x_\alpha)$ in a vector lattice $X$ is said to uo-converge to $x$ if $|x_\alpha-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Ces\`aro means in Banach lattices. In particular, we establish an intrinsic version of Koml\'os' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25]. Archive classification: math.FA Remarks: 45 pages Submitted from: foivos(a)ryerson.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.07914 or http://arXiv.org/abs/1509.07914

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Abstract of a paper by Gleb Sirotkin and Ben Wallis
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "Almost-invariant and essentially-invariant halfspaces" by Gleb Sirotkin and Ben Wallis. Abstract: In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if $X$ is an infinite-dimensional complex Banach space then every operator $T\in\mathcal{L}(X)$ admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well. Archive classification: math.FA Mathematics Subject Classification: 15A03, 15A18, 15A60, 47L10, 47A10, 47A11, 47A15 Remarks: 11 pages. Keywords: functional analysis, Banach spaces, surjectivity The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.07428 or http://arXiv.org/abs/1509.07428

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Abstract of a paper by Grigory Ivanov
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "Hypomonotonicity of the normal cone and proximal smoothness" by Grigory Ivanov. Abstract: In this paper we study the properties of the normal cone to the proximally smooth set. We give the complete characterization of the proximally smooth set through the monotony properties of its normal cone in an arbitrary uniformly convex and uniformly smooth Banach space. We give the exact bounds for right-hand side in the monotonicity inequality for normal cone in terms of the moduli of smoothness and convexity of a Banach space. Archive classification: math.FA Submitted from: grigory.ivanov(a)phystech.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.06795 or http://arXiv.org/abs/1509.06795

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Abstract of a paper by Hana Krulisova
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "Quantification of Pe\l czy\'nski's property (V)" by Hana Krulisova. Abstract: A Banach space $X$ has Pe\l czy\' nski's property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\to Y$ is weakly compact. In 1962, Aleksander Pe\l czy\' nski showed that $C(K)$ spaces for a compact Hausdorff space $K$ enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about $C(K)$ spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces. Archive classification: math.FA Remarks: 19 pages Submitted from: krulisova(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.06610 or http://arXiv.org/abs/1509.06610

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Abstract of a paper by Zhe-Ming Zheng and Hui-Sheng Ding
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "A note on closedness of the sum of two closed subspaces in a Banach space" by Zhe-Ming Zheng and Hui-Sheng Ding. Abstract: Let $X$ be a Banach space, and $M,N$ be two closed subspaces of $X$. We present several necessary and sufficient conditions for the closedness of $M+N$ ($M+N$ is not necessarily direct sum). Archive classification: math.FA Submitted from: dinghs(a)mail.ustc.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.06445 or http://arXiv.org/abs/1509.06445

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Abstract of a paper by Silouanos Brazitikos
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "Brascamp-Lieb inequality and quantitative versions of Helly's theorem" by Silouanos Brazitikos. Abstract: We provide a number of new quantitative versions of Helly's theorem. For example, we show that for every family $\{P_i:i\in I\}$ of closed half-spaces $$P_i=\{ x\in {\mathbb R}^n:\langle x,w_i\rangle \leq 1\}$$ in ${\mathbb R}^n$ such that $P=\bigcap_{i\in I}P_i$ has positive volume, there exist $s\leq \alpha n$ and $i_1,\ldots , i_s\in I$ such that $$|P_{i_1}\cap\cdots\cap P_{i_s}|\leq (Cn)^n\,|P|,$$ where $\alpha , C>0$ are absolute constants. These results complement and improve previous work of B\'{a}r\'{a}ny-Katchalski-Pach and Nasz\'{o}di. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp-Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality. Archive classification: math.FA Mathematics Subject Classification: 26D15 Submitted from: silouanb(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.05783 or http://arXiv.org/abs/1509.05783

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Abstract of a paper by Gonzalo Martinez-Cervantes
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "On weakly Radon-Nikod\'ym compact spaces" by Gonzalo Martinez-Cervantes. Abstract: A compact space is said to be weakly Radon-Nikod\'ym if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of $\ell_1$. In this work we provide an example of a continuous image of a Radon-Nikod\'ym compact space which is not weakly Radon-Nikod\'ym. Moreover, we define a superclass of the continuous images of weakly Radon-Nikod\'ym compact spaces and study its relation with Corson compacta and weakly Radon-Nikod\'ym compacta. Archive classification: math.FA math.GN Mathematics Subject Classification: 46B22, 46B50, 54G20 Submitted from: gonzalo.martinez2(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.05324 or http://arXiv.org/abs/1509.05324

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Abstract of a paper by Tatiana Shulman
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "On subspaces of invariant vectors" by Tatiana Shulman. Abstract: Let $X_{\pi}$ be the subspace of fixed vectors for a uniformly bounded representation $\pi$ of a group $G$ on a Banach space $X$. We study the problem of the existence and uniqueness of a subspace $Y$ that complements $X_{\pi}$ in $X$. Similar questions for $G$-invariant complement to $X_{\pi}$ are considered. We prove that every non-amenable discrete group $G$ has a representation with non-complemented $X_{\pi}$ and find some conditions that provide an $G$-invariant complement. A special attention is given to representations on $C(K)$ that arise from an action of $G$ on a metric compact $K$. Archive classification: math.FA Mathematics Subject Classification: 22A25, 46B99, 22D25 Submitted from: tatiana_shulman(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.05263 or http://arXiv.org/abs/1509.05263

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