Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Thomas Delzant, Vilmos Komornik
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Representations of dual spaces” by Thomas Delzant<https://arxiv.org/search/math?searchtype=author&query=Delzant%2C+T>, Vilmos Komornik<https://arxiv.org/search/math?searchtype=author&query=Komornik%2C+V>. Abstract: We give a nonlinear representation of the duals for a class of Banach spaces. This leads to classroom-friendly proofs of the classical representation theorems $H'=H$ and $(L^p)'=L^q$. Our proofs extend to a family of Orlicz spaces, and yield as an unexpected byproduct a version of the Helly-Hahn-Banach theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.06811

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Abstract of a paper by Joscha Prochno
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Embeddings of Orlicz-Lorentz spaces into $L_1$” by Joscha Prochno<https://arxiv.org/search/math?searchtype=author&query=Prochno%2C+J>. Abstract: In this article, we show that Orlicz-Lorentz spaces $\ell^n_{M,a}$, $n\in\mathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\|\cdot\|_{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $d^n(a,p)$. Our approach is based on combinatorial averaging techniques and we prove a new result of independent interest that relates suitable averages with Orlicz-Lorentz norms. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.05043

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Abstract of a paper by Niels Jakob Laustsen, Jared T. White
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Subspaces that can and cannot be the kernel of a bounded operator on a Banach space” by Niels Jakob Laustsen<https://arxiv.org/search/math?searchtype=author&query=Laustsen%2C+N+J>, Jared T. White<https://arxiv.org/search/math?searchtype=author&query=White%2C+J+T>. Abstract: Given a Banach space $E$, we ask which closed subspaces may be realised as the kernel of a bounded operator $E \rightarrow E$. We prove some positive results which imply in particular that when $E$ is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space $E$ which contains a closed subspace that cannot be realized as the kernel of any bounded operator on $E$. This implies that the Banach algebra $\mathcal{B}(E)$ of bounded operators on $E$ fails to be weak*-topologically left Noetherian. The Banach space $E$ that we use is the dual of Wark's non-separable, reflexive Banach space with few operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.01677

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Abstract of a paper by M.D. Acosta, M. Soleimani-Mourchehkhorti
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Stability results of properties related to the Bishop-Phelps-Bollobás property for operators” by M.D. Acosta<https://arxiv.org/search/math?searchtype=author&query=Acosta%2C+M+D>, M. Soleimani-Mourchehkhorti<https://arxiv.org/search/math?searchtype=author&query=Soleimani-Mourchehkho…>. Abstract: We prove that the class of Banach spaces $Y$ such that the pair $(\ell_1, Y)$ has the Bishop-Phelps-Bollobás property for operators is stable under finite products when the norm of the product is given by an absolute norm. We also provide examples showing that previous stability results obtained for that property are optimal. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.01677

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Abstract of a paper by Aziz Azizov, Vladimir Chilin, Semyon Litvinov
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Ergodic theorems in Banach ideals of compact operators” by Aziz Azizov<https://arxiv.org/search/math?searchtype=author&query=Azizov%2C+A>, Vladimir Chilin<https://arxiv.org/search/math?searchtype=author&query=Chilin%2C+V>, Semyon Litvinov<https://arxiv.org/search/math?searchtype=author&query=Litvinov%2C+S>. Abstract: Let $\mathcal H$ be a complex infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a fully symmetric sequence space. If $\{s_n(x)\}_{n=1}^\infty$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{s_n(x)\}\subset E\}$ with $\|x\|_{\mathcal C_E}=\|\{s_n(x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. We show that the averages $A_n(T)(x)=\frac1{n+1}\sum\limits_{k = 0}^n T^k(x)$ converge uniformly in $\mathcal C_E$ for any positive Dunford-Schwartz operator $T$ and $x\in\mathcal C_E$. Besides, if $x\in\mathcal B(\mathcal H)\setminus\mathcal K(\mathcal H)$, there exists a Hermitian Dunford-Schwartz operator $T$ such that the sequence $\{A_n(T)(x)\}$ does not converge uniformly. We also show that the averages $A_n(T)$ converge strongly in $(\mathcal C_E,\|\cdot\|_{\mathcal C_E})$ if and only if $E$ is separable and $E\neq l^1$, as sets. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.00759

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Abstract of a paper by Rafael Chiclana, Miguel Martin
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “The Bishop--Phelps--Bollobás property for Lipschitz maps” by Rafael Chiclana<https://arxiv.org/search/math?searchtype=author&query=Chiclana%2C+R>, Miguel Martin<https://arxiv.org/search/math?searchtype=author&query=Martin%2C+M>. Abstract: In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility to make a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness are needed. Next, we show that if $M$ is a uniformly Gromov concave pointed metric space (i.e.\ the molecules of $M$ form a set of uniformly strongly exposed points), then $(M,Y)$ has the Lip-BPB property for every Banach space $Y$. We further prove that this is the case of finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from $(M,\mathbb{R})$ to some Banach spaces $Y$, the relationship with absolute sums, and some results only valid for compact Lipschitz maps, are also discussed. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1901.02956

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Abstract of a paper by Ivan Yaroslavtsev
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “On strongly orthogonal martingales in UMD Banach spaces” by Ivan Yaroslavtsev<https://arxiv.org/search/math?searchtype=author&query=Yaroslavtsev%2C+I>. Abstract: In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1<p<\infty$ \[ \mathbb E \|N_t\|^p \leq \chi_{p, X}^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0, \] with the sharp constant $\chi_{p, X}$ being the norm of a decoupling-type martingale transform and being within the range \[ \max\Bigl\{\sqrt{\beta_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr\} \leq \max\{\beta_{p, X}^{\gamma,+}, \beta_{p, X}^{\gamma, -}\} \leq \chi_{p, X} \leq \min\{\beta_{p, X}, \hbar_{p,X}\}, \] where $\beta_{p, X}$ is the UMD$_p$ constant of $X$, $\hbar_{p, X}$ is the norm of the Hilbert transform on $L^p(\mathbb R; X)$, and $\beta_{p, X}^{\gamma,+}$ and $ \beta_{p, X}^{\gamma, -}$ are the Gaussian decoupling constants. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.08049

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Abstract of a paper by March T. Boedihardjo
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Bounded representations on $l^p$” by March T. Boedihardjo<https://arxiv.org/search/math?searchtype=author&query=Boedihardjo%2C+M+T>. Abstract: We show that (i) every bounded unital representation of an amenable group $G$ on $l^{p}$, $1<p<\infty$, is a direct summand of a representation that is approximately similar to the left regular representation of $G$ on $l^{p}$ and that (ii) if $\rho$ is a unital representation of a unital $C^{*}$-algebra $\mathcal{A}$ on $l^{p}$, $1<p<\infty$, $p\neq 2$, then $\rho$ satisfies a compactness property and $\mathcal{A}/\text{ker }\rho$ is residually finite dimensional. As a consequence, a separable unital $C^{*}$-algebra $\mathcal{A}$ is isomorphic to a subalgebra of $B(l^{p})$, $1<p<\infty$, $p\neq 2$, if and only if $\mathcal{A}$ is residually finite dimensional. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.11165

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Abstract of a paper by Ioakeim Ampatzoglou
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “On the non-embedding of $\ell_1$ in the James Tree Space” by Ioakeim Ampatzoglou<https://arxiv.org/search/math?searchtype=author&query=Ampatzoglou%2C+I>. Abstract: James Tree Space ($\mathcal{JT}$), introduced by R. James, is the first Banach space constructed having non-separable conjugate and not containing $\ell^1$. James actually proved that every infinite dimensional subspace of $\mathcal{JT}$ contains a Hilbert space, which implies the $\ell^1$ non-embedding. In this expository article, we present a direct proof of the $\ell^1$ non-embedding, using Rosenthal's $\ell^1$- Theorem and some measure theoretic arguments, namely Riesz's Representation Theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.07825

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Abstract of a paper by Karol Leśnik, Jakub Tomaszewski
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Pointwise multipliers of Musielak--Orlicz spaces and factorization” by Karol Leśnik<https://arxiv.org/search/math?searchtype=author&query=Le%C5%9Bnik%2C+K>, Jakub Tomaszewski<https://arxiv.org/search/math?searchtype=author&query=Tomaszewski%2C+J>. Abstract: We prove that the space of pointwise multipliers between two distinct Musielak--Orlicz spaces is another Musielak-Orlicz space and the function defining it is given by an appropriately generalized Legendre transform. In particular, we obtain characterization of pointwise multipliers between Nakano spaces. We also discuss factorization problem for Musielak-Orlicz spaces and exhibit some differences between Orlicz and Musielak-Orlicz cases. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.05887

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