Banach January 2020

banach@mathdept.okstate.edu

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Announcement of a Postdoc Position at University of Memphis
by Bentuo Zheng (bzheng) 24 Mar '20

24 Mar '20

Dear All, There will be a two-year postdoc position at the Department of Mathematical Sciences of the University of Memphis which may be renewable for the third year. Please find the details in the attached ad. BR, Bentuo Zheng

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Announcement of a paper by M. Brixey, R.M. Causey, P. Frankart
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper Uniform subsequential estimates on weakly null sequences” by M. Brixey<https://arxiv.org/search/math?searchtype=author&query=Brixey%2C+M>, R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M>, P. Frankart<https://arxiv.org/search/math?searchtype=author&query=Frankart%2C+P>. Abstract: We provide a generalization of two results of Knaust and Odell from \cite{KO2} and \cite{KO}. We prove that if $X$ is a Banach space and $(g_n)_{n=1}^\infty$ is a right dominant Schauder basis such that every normalized, weakly null sequence in $X$ admits a subsequence dominated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists a constant $C$ such that every normalized, weakly null sequence in $X$ admits a subsequence $C$-dominated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove that if every spreading model generated by a normalized, weakly null sequence in $X$ is dominated by some spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists $C$ such that every spreading model generated by a normalized, weakly null sequence in $X$ is $C$-dominated by every spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove a single, ordinal-quantified result which unifies and interpolates between these two results. https://arxiv.org/abs/1912.13443

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Announcement of a paper by R.M. Causey
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “$Sz(\cdot)\leqslant ω^ξ$ is rarely a three space property” by R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M>. Abstract: We prove that for any non-zero, countable ordinal $\xi$ which is not additively indecomposable, the property of having Szlenk index not exceeding $\omega^\xi$ is not a three space property. This complements a result of Brooker and Lancien, which states that if $\xi$ is additively indecomposable, then having Szlenk index not exceeding $\omega^\xi$ is a three space property. https://arxiv.org/abs/1912.13429

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Announcement of a paper by Joanna Garbulińska-Wegrzyn, Wieslaw Kubiś
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “A note on universal operators between separable Banach spaces” by Joanna Garbulińska-Wegrzyn<https://arxiv.org/search/math?searchtype=author&query=Garbuli%C5%84ska-Wegr…>, Wieslaw Kubiś<https://arxiv.org/search/math?searchtype=author&query=Kubi%C5%9B%2C+W>. Abstract: We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator $\Omega$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $\Omega$ is generic in the sense of a natural infinite game. https://arxiv.org/abs/1912.13312

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Announcement of a paper by Trond A. Abrahamsen, Petr Hájek, Stanimir Troyanski
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “Almost square dual Banach spaces” by Trond A. Abrahamsen<https://arxiv.org/search/math?searchtype=author&query=Abrahamsen%2C+T+A>, Petr Hájek<https://arxiv.org/search/math?searchtype=author&query=H%C3%A1jek%2C+P>, Stanimir Troyanski<https://arxiv.org/search/math?searchtype=author&query=Troyanski%2C+S>. Abstract: We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals. https://arxiv.org/abs/1912.12519

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Announcement of a paper by Nigel J. Kalton, Emiel Lorist, Lutz Weis
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “Euclidean structures and operator theory in Banach spaces” by Nigel J. Kalton<https://arxiv.org/search/math?searchtype=author&query=Kalton%2C+N+J>, Emiel Lorist<https://arxiv.org/search/math?searchtype=author&query=Lorist%2C+E>, Lutz Weis<https://arxiv.org/search/math?searchtype=author&query=Weis%2C+L>. Abstract: We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators $\Gamma$ on a Hilbert space. Our assumption on $\Gamma$ is expressed in terms of $\alpha$-boundedness for a Euclidean structure $\alpha$ on the underlying Banach space $X$. This notion is originally motivated by $\mathcal{R}$- or $\gamma$-boundedness of sets of operators, but for example any operator ideal from the Euclidean space $\ell^2_n$ to $X$ (like the $\gamma$-radonifying or the $2$-summing operator ideal) defines such a structure. Therefore our method is quite general and flexible and allows to unify the approach to seemingly unrelated theorems. Conversely we show that $\Gamma$ has to be $\alpha$-bounded for some Euclidean structure $\alpha$ for it to be representable on a Hilbert space. By choosing the Euclidean structure $\alpha$ accordingly we get a unified and more general approach to classical factorization and extension theorems. Furthermore we use these Euclidean structures to build vector-valued function spaces and define an interpolation method based on these spaces, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem we prove a quite general transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We define generalizations of the classical square function estimates in $L^p$-spaces and establish via the $H^\infty$-calculus a version of Littlewood-Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our results for sectorial operators lead to some sophisticated counterexamples. https://arxiv.org/abs/1912.09347

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Announcement of a paper by Ginés López-Pérez, Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “$L$-orthogonality, octahedrality and Daugavet property in Banach spaces” by Ginés López-Pérez<https://arxiv.org/search/math?searchtype=author&query=L%C3%B3pez-P%C3%A9rez…>, Abraham Rueda Zoca<https://arxiv.org/search/math?searchtype=author&query=Zoca%2C+A+R>. Abstract: We prove that the abundance of almost $L$-orthogonal vectors in a Banach space $X$ (almost Daugavet property) implies the abundance of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$. In fact, we get that a Banach space $X$ verifies the Daugavet property if, and only if, the set of vectors in $X^{**}$ being $L$-orthogonal to $X$ is weak-star dense in $X^{**}$. In contrast with the separable case, we prove that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the existence of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$, which shows that the answer to an environment question in [7] is negative. Also, in contrast with the separable case, we obtain that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the abundance of almost $L$-orthogonal vectors in Banach space $X$ (almost Daugavet property), which solves an open question in [21]. Some consequences on Daugavet property in the setting of $L$-embedded spaces are also obtained. https://arxiv.org/abs/1912.09039

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Announcement of a paper by Tommaso Russo, Jacopo Somaglia
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “Overcomplete sets in Banach spaces” by Tommaso Russo<https://arxiv.org/search/math?searchtype=author&query=Russo%2C+T>, Jacopo Somaglia<https://arxiv.org/search/math?searchtype=author&query=Somaglia%2C+J>. Abstract: We introduce and study the notion of overcomplete set in a Banach space, that subsumes and extends the classical concept of overcomplete sequence in a (separable) Banach space. We give existence and non-existence results of overcomplete sets for a wide class of (non-separable) Banach spaces and we study to which extent properties of overcomplete sequences are retained by every overcomplete set. https://arxiv.org/abs/1912.08690

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Announcement of a paper by Claudia Correa
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “On the $c_0$-extension property” by Claudia Correa<https://arxiv.org/search/math?searchtype=author&query=Correa%2C+C>. Abstract: In this work we investigate the c_0-extension property. This property generalizes Sobczyk's theorem in the context of nonseparable Banach spaces. We prove that a sufficient condition for a Banach space to have this property is that its closed dual unit ball is weak-star monolithic. We also present several results about the c_0-extension property in the context of C(K) Banach spaces. An interesting result in the realm of C(K) spaces is that the existence of a Corson compactum K such that C(K) does not have the c_0-extension property is independent from the axioms of ZFC. https://arxiv.org/abs/1912.08564

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Announcement of a paper by Fernando Albiac, José L. Ansorena, Przemysław Wojtaszczyk
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “On certain subspaces of $\ell_p$ for $0<p\le 1$ and their applications to conditional quasi-greedy bases in $p$-Banach spaces” by Fernando Albiac<https://arxiv.org/search/math?searchtype=author&query=Albiac%2C+F>, José L. Ansorena<https://arxiv.org/search/math?searchtype=author&query=Ansorena%2C+J+L>, Przemysław Wojtaszczyk<https://arxiv.org/search/math?searchtype=author&query=Wojtaszczyk%2C+P>. Abstract: We construct for each $0<p\le 1$ an infinite collection of subspaces of $\ell_p$ that extend the example from [J. Lindenstrauss, On a certain subspace of $\ell_{1}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 539-542] of a subspace of $\ell_{1}$ with no unconditional basis. The structure of this new class of $p$-Banach spaces is analyzed and some applications to the general theory of $\mathcal{L}_{p}$-spaces for $0<p<1$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in $p$-Banach spaces for $p<1$. Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces $\ell_{p}$ for $p\le 1$ and the careful examination of the conditionality constants of the "natural basis" of these spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1912.08449

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Banach January 2020