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Abstract of a paper by Richard J. Smith, Stanimir Troyanski
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Approximation of norms on Banach spaces” by Richard J. Smith<https://arxiv.org/search?searchtype=author&query=Smith%2C+R+J>, Stanimir Troyanski<https://arxiv.org/search?searchtype=author&query=Troyanski%2C+S>. Abstract: Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^{\infty}$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discrete' Lorentz spaces $d(w, 1,\gamma)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $\alpha$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $\alpha$, such that every equivalent norm on $C(K)$ can be approximated as above. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.05660

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Abstract of a paper by Stephen J. Dilworth, Van Kien Nguyen
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Greedy Algorithms and Kolmogorov Widths in Banach Spaces” by Stephen J. Dilworth<https://arxiv.org/find/math/1/au:+Dilworth_S/0/1/0/all/0/1>, Van Kien Nguyen<https://arxiv.org/search?searchtype=author&query=Nguyen%2C+V+K>. Abstract: Let $X$ be a Banach space and $K$ be a compact subset in $X$. We consider a greedy algorithm for finding $n$-dimensional subspace $V_n\subset X$ which can be used to approximate the elements of $K$. We are interested in how well the space $V_n$ approximates the elements of $K$. For this purpose we compare the greedy algorithm with the Kolmogorov, width which is the best possible error one can approximate $K$ by $n$−dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (SIAM J. Math. Anal. (2011)), DeVore et al. (Constr. Approx. (2013)) and Wojtaszczyk (J. Math. Anal. Appl. (2015)). The purpose of the present paper is to continue this line. We shall show that under some additional assumptions the results in the above-mentioned papers can be improved. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.03935

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Abstract of a paper by Richard J. Smith
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Topology, isomorphic smoothness and polyhedrality in Banach spaces” by Richard J. Smith<https://arxiv.org/search?searchtype=author&query=Smith%2C+R+J>. Abstract: In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called $w*$-locally relatively compact sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.02899

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Abstract of a paper by Rainis Haller, Johann Langemets, Vegard Lima, Rihhard Nadel
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Symmetric strong diameter two property” by Rainis Haller<https://arxiv.org/search?searchtype=author&query=Haller%2C+R>, Johann Langemets<https://arxiv.org/search?searchtype=author&query=Langemets%2C+J>, Vegard Lima<https://arxiv.org/search?searchtype=author&query=Lima%2C+V>, Rihhard Nadel<https://arxiv.org/search?searchtype=author&query=Nadel%2C+R>. Abstract: We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and P\~oldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2. We show that the symmetric strong diameter two property is only preserved by $\ell_{\infty}$-sums, and working with weak star slices we show that $\Lip_0(M)$ have the weak star version of the property for several classes of metric spaces $M$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.01705

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Abstract of a paper by Richard Lechner
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Subsymmetric weak$^*$ Schauder bases and factorization of the identity” by Richard Lechner<https://arxiv.org/find/math/1/au:+Lechner_R/0/1/0/all/0/1>. Abstract: Let $X^*$ denote a Banach space with a subsymmetric weak$^*$ Schauder basis satisfying condition. We show that for any operator $T: X^*\rightarrow X^*$ either $T(X^*)$ or $(I-T)(X^*)$ contains a subspace that is isomorphic to $X^*$ and complemented in $X^*$. Moreover, we prove that $\ell_p(X^*), 1\leq p\leq\infty$ is primary. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.01372

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Conference Announcement
by Bentuo Zheng (bzheng) 16 Apr '18

16 Apr '18

Complex Approximation and Bounded Holomorphic Functions Readers of the Banach Bulletin Board may be interested in an upcoming meeting on "Complex Approximation and Bounded Holomorphic Functions,” to be held near Trento, Italy, on Monday – Friday, October 1 – 5, 2018. The Conference organizers are R. M. Aron (Kent-OH, USA), S. Charpentier (Marseilles), M. Maestre (Valencia), and V. Nestoridis (Athens). A basic theme is to study a few areas in several and infinitely many complex variables. Principal among these areas are: (1) Polynomial and Rational Approximation, extendibility of holomorphic functions, and generic nature of non-extendible functions, and (2) Banach algebras of bounded holomorphic functions in one, several, and infinitely many variables. Topics of special interest include cluster value theorems (i.e. weak forms of the Corona Theorem), relations with Dirichlet series, and the study of Gleason parts in the maximal ideal spaces of such algebras. The Conference has received partial support from CIRM, which will cover the cost of room and board for two of the five days of the meeting. Unfortunately, no further financial aid is available. At the moment, there are several places still available, to an upper limit of 40 participants. Interested mathematicians are asked to kindly contact R. M. Aron: aron(a)math.kent.edu<mailto:aron@math.kent.edu> .

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Abstract of a paper by Mark Veraar, Ivan Yaroslavtsev
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “Pointwise properties of martingales with values in Banach function spaces” by Mark Veraar<https://arxiv.org/find/math/1/au:+Veraar_M/0/1/0/all/0/1>, Ivan Yaroslavtsev<https://arxiv.org/find/math/1/au:+Yaroslavtsev_I/0/1/0/all/0/1>. Abstract: In this paper we consider local martingales with values in a UMD Banach function space. We prove that such martingales have a version which is a martingale field. Moreover, a new Burkholder--Davis--Gundy type inequality is obtained. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.11063

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Abstract of a paper by Manor Mendel
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “A Simple proof of Johnson-Lindenstrauss extension” by Manor Mendel<https://arxiv.org/find/math/1/au:+Mendel_M/0/1/0/all/0/1>. Abstract: Johnson and Lindenstrauss proved that any Lipschitz mapping from $n$-point metric space into Hilbert space can be extended while losing at most a factor of $O(\sqrt{\log n})$ in the Lipschitz constant. We present a variation of their argument that avoids dimension reduction and Kirszbraun theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.03606

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Abstract of a paper by R.M. Causey, K. Navoyan
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “$xi$-completely continuous operators and $\xi$-Schur Banach spaces” by R.M. Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>, K. Navoyan<https://arxiv.org/find/math/1/au:+Navoyan_K/0/1/0/all/0/1>. Abstract: For each ordinal $0\leq\xi\leq\omega_1$, we introduce the notion of a $\xi$-completely continuous operator and prove that for each ordinal $0<\xi<\omega_1$, the class $\mathcal{B}_{\xi}$of $\xi$-completely continuous operators is a closed, injective operator ideal which is not surjective, symmetric, or idempotent. We prove that for distinct $0\leq\xi, \zeta\leq\omega_1$, the classes of $\xi$-completely continuous operators and $\zeta$-completely continuous operators are distinct. We also introduce an ordinal rank $\nu$ for operators such that $\nu(A)=\omega_1$ if and only if $A$ is completely continuous, and otherwise $\nu(A)$ is the minimum countable ordinal such that $A$ fails to be $\xi$-completely continuous. We show that there exists an operator $A$ such that $\nu(A)=\xi$ if and only if $1\leq\xi\leq\omega_1$, and there exists a Banach space $X$ such that $\nu(I_X)=\xi$ if and only if there exists an ordinal $\gamma\leq\omega_1$ such that $\xi=\omega^{\gamma}$. Finally, prove that for every $0<\xi<\omega_1$, the class $\{A\in\mathcal{L}:\nu(A)\geq\xi\}$ is $\Pi_1^1$-complete in $\mathcal{L}$, the coding of all operators between separable Banach spaces. This is in contrast to the class $\mathcal{B}\cap\mathcal{L}$, which is $\Pi_2^1$-complete in $\mathcal{L}$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.09343

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Abstract of a paper by Fernando Albiac, Jose L. Ansorena, Stephen Dilworth, Denka Kutzarova
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “Building highly conditional almost greedy and quasi-greedy bases in Banach spaces” by Fernando Albiac<https://arxiv.org/find/math/1/au:+Albiac_F/0/1/0/all/0/1>, Jose L. Ansorena<https://arxiv.org/find/math/1/au:+Ansorena_J/0/1/0/all/0/1>, Stephen Dilworth<https://arxiv.org/find/math/1/au:+Dilworth_S/0/1/0/all/0/1>, Denka Kutzarova<https://arxiv.org/find/math/1/au:+Kutzarova_D/0/1/0/all/0/1>. Abstract: It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_m[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_m[\mathcal{B}=O(\log m)$ and that if the reverse inequality $\logm=O(k_m[\mathcal{B}) holds then $\mathbb{X}$ is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate $k_m[\mathcal{B}=O(\log m)^{1-\epsilon}$ for some $\epsilon>0$. However, in the existing literature one finds very few instances of spaces possessing quasi-greedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed in [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases $\mathcal{B}$ with $k_m[\mathcal{B}=O(\log m)$ and superreflexive classical Banach spaces having for every $\epsilon>0$ quasi-greedy bases $\mathcal{B}$ with $k_m[\mathcal{B}=O(\log m)^{1-\epsilon}$. Moreover, in most cases those bases will be almost greedy. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.08351

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