April 2018 - Banach - mathdept.okstate.edu

Informal Analysis Seminar February 24-25, 2018
by Artem Zvavitch 05 Sep '19

05 Sep '19

Dear Colleagues, The Analysis group at Kent State University is happy to announce a meeting of the Informal Analysis Seminar, which will be held at the Department of Mathematical Sciences at Kent State University, February 24-25. The seminar will feature plenary speakers Robert Connelly (Cornell University), and Peter Sternberg (Indiana University) Each speaker will deliver a four hour lecture series designed to be accessible for graduate students. Funding is available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Posters can be submitted electronically in PDF format. Further information, and an online registration form, can be found online http://www.math.kent.edu/informal We encourage you to register as soon as possible, but to receive support and/or help with hotel reservation, please register before January 29, 2018. Finally, please feel free to forward this email to any colleagues or students who you think may be interested in attending. Best regards, The Kent State Analysis Group

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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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Abstract of a paper by Osamu Hatori
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “Isometries on Banach algebras of C(Y)-valued maps” by Osamu Hatori<https://arxiv.org/find/math/1/au:+Hatori_O/0/1/0/all/0/1>. Abstract: We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of natural $C(Y)$-valuezations that take values in unital commutative $C^*$-algebras. A precise proof of a theorem of Jarosz \cite{ja} is exhibited. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.08236

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Abstract of a paper by Marek Balcerzak, Michał Popławski, Artur Wachowicz
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “Ideal convergent subseries in Banach spaces” by Marek Balcerzak<https://arxiv.org/find/math/1/au:+Balcerzak_M/0/1/0/all/0/1>, Michał Popławski<https://arxiv.org/find/math/1/au:+Poplawski_M/0/1/0/all/0/1>, Artur Wachowicz<https://arxiv.org/find/math/1/au:+Wachowicz_A/0/1/0/all/0/1>. Abstract: Assume that $\mathcal{I}$ is an ideal on $\mathbb{N}$, and $\sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(\mathcal{I}):=\{t\in\{0,1\}^{\mathbb{N}}: \sum_n t(n)x_n\textrm{is} \mathcal{I}-\textrm{convergent}\}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$ is not unconditionally convergent, and we deduce that $A(\mathcal{I})$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure λ on {0,1}ℕ is considered. If $\mathcal{I}$ is analytic or coanalytic, and $\sum_n x_n$ is $\mathcal{I}$-divergent, then $\lambda(A(\mathcal{I}))=0$ which extends the theorem of Dindo\v{s}, \v{S}al\'at and Toma. Generalizing one of their examples, we show that, for every ideal $\mathcal{I}$ on $\mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $\lambda(A(Fin))=0$ and $\lambda(A(\mathcal{I}))=1$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.03699

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Abstract of a paper by Stephen J. Dilworth, Denka Kutzarova, Vladimir Temlyakov, Ben Wallis
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “Weight-almost greedy bases” by Stephen J. 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>, Vladimir Temlyakov<https://arxiv.org/find/math/1/au:+Temlyakov_V/0/1/0/all/0/1>, Ben Wallis<https://arxiv.org/find/math/1/au:+Wallis_B/0/1/0/all/0/1>. Abstract: We introduce the notion of a \textit{weight-almost greedy} basis and show that a basis for a real Banach space is w-almost greedy if and only if it is both quasi-greedy and w-democratic. We also introduce the notion of \textit{weight-semi-greedy} basis and show that a w-almost greedy basis is w-semi-greedy and that the converse holds if the Banach space has finite cotype. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.02932

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Abstract of a paper by Ioana Ghenciu
by Bentuo Zheng (bzheng) 01 Apr '18

01 Apr '18

This is an announcement for the paper “The p-Gelfand Phillips Property in Spaces of Operators and Dunford-Pettis like sets” by Ioana Ghenciu<https://arxiv.org/find/math/1/au:+Ghenciu_I/0/1/0/all/0/1>. Abstract: The $p$-Gelfand Phillips property $(1\leq p<\infty)$ is studied in spaces of operators. Dunford - Pettis type like sets are studied in Banach spaces. We discuss Banach spaces $X$ with the property that every $p$-convergent operator $T: X\rightarrow Y$ is weakly compact, for every Banach space $Y$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.00351

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