May 2016 - Banach - mathdept.okstate.edu

INFORMAL ANALYSIS SEMINAR DEDICATED TO THE WORK OF JOE DIESTEL (March 20-21, 2010)
by Artem Zvavitch 20 Aug '16

20 Aug '16

Dear Friends, On Saturday - Sunday, March 20-21, 2010, (best arrival date Friday, March 19/ Departure Monday, March 22) the Department of Mathematical Sciences at Kent State University will be famous but still very informal. We are happy to announce an: INFORMAL ANALYSIS SEMINAR DEDICATED TO THE WORK OF JOE DIESTEL The following people are among speakers: Peter Casazza, Hans Jarchow, Alexander Koldobsky, Pepe Orihuela, Olek Pelczynski, Thomas Schlumprecht, Lutz Weis. It would be great if you could visit Kent State and participate in the seminar! The conference will have a registration fee of $200 WHICH INCLUDES: pick up/drop off from the airport, THREE DAY STAY in the Microtel hotel at Streetsboro, Saturday/Sunday lunches/dinners to be provided at the department. If you plan to stay fewer than 3 nights or prefer to make your own accommodation arrangements, please reduce your registration fee by $45 per day that you will not use our hotel arrangements. If possible, please send a check for your registration fee, made out to the Department of Mathematical Sciences, Kent State University. The check should be mailed to Virginia Wright, Department of Mathematical Sciences, Kent State University, Kent, OH, 44242. The fee can be also be paid during the registration (check/cash). The conference is supported by NSF; The Department of Mathematical Sciences and Kent State University. Depending on funds availability the fee may be waived for young researchers and people without available funding. Please, write as soon as possible to zvavitch(a)math.kent.edu May we ask you to respond as soon as February 19 (zvavitch(a)math.kent.edu) so that we can gauge the need for housing, lecture room(s), etc. Best Regards, Analysis group at Kent State!

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Abstract of a paper by Jan Rozendaal
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Fourier multiplier theorems involving type and cotype” by Jan Rozendaal. Abstract: In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^{d}\to\mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's and is reasonably well-understood. Far less is known for $p<q$. In the scalar case one can deduce results for $p<q$ from the case $p=q$ and additional arguments such as interpolation and Sobolev embedding techniques. However, in the vector-valued case this leads to restrictions both on the smoothness of the multiplier and the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces, and that $m$ satisfies a smoothness condition. For $p<q$ it turns out that the notions of type and cotype and other geometric conditions on $X$ and $Y$ are important to study Fourier multipliers. Moreover, we obtain boundedness results for $T_{m}$ without any smoothness properties of $m$. Under additional smoothness conditions we show that boundedness results can be extrapolated to other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ remains constant. Here we even extend classical results in the scalar case. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.09340

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Abstract of a paper by Asuman G. Aksoy and Qidi Peng
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “On a Theorem of S. N. Bernstein for Banach Spaces” by Asuman G. Aksoy and Qidi Peng. Abstract: This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We prove that if $X$ is an infinite-dimensional Banach space and $(Y_n)$ is a nested sequence of subspaces of $X$ such that $Y_n\subset Y_{n+1}$ and $\bar{Y_n}\subset Y_{n+1}$ for any $n\in\mathbb{N}$ and if $(d_n)$ be a decreasing sequence of positive numbers tending to 0, then for any $0<c\leq 1$ there exists $x_c\in X$ such that the distance $\rho (x_c, Y_n)$ from $x_c$ to $Y_n$ satisfies $$cd_n\leq\rho(x_c, Y_n)\leq 4c d_n$$. We prove the above by first improving Borodin's result \cite{Borodin} for Banach spaces by weakening the condition on the sequence $(d_n)$. Lastly, we compare subsequences $(d(\phi_n))$ under different choices of $(\phi_n)$ and examine their effects on approximation. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.04592

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Abstract of a paper by S. Ferrari, J. Orihuela and M. Raja
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Generalized metric properties of spheres and renorming of normed spaces” by S. Ferrari, J. Orihuela and M. Raja. Abstract: We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove that there exists a dual norm such that its unit sphere is a Moore space for the weak$^*$topology (has a $G_\delta$-diagonal for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively). The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.08175

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Abstract of a paper by Mohammad Imdad, Rqeeb Gubran and Md Ahmadullah
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Using an implicit function to prove common fixed point theorems” by Mohammad Imdad, Rqeeb Gubran and Md Ahmadullah. Abstract: We observe that, Theorem 2 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105) is not correct in its present form. In this paper, by correcting and enriching aforementioned theorem, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike many other cases, our main results deduce a nonlinear order-theoretic version of a well-known fixed point theorem (proved for quasi-contraction) due to $\'{C}iri\'{c}$ (Proc. Amer. Math. Soc. (54) 267-273, 1974). Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105). The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.05743

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Abstract of a paper by Chafiq Benhida, Kamila Klis-Garlicka and Marek Ptak
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Skew-symmetric operators and reflexivity” by Chafiq Benhida, Kamila Klis-Garlicka and Marek Ptak. Abstract: In contrast to the subspaces of all C-symmetric operators, we show that the subspaces of all skew-C symmetric operators are reflexive and even hyperreflexive with the constant $\kappa(\C^s)\leqslant 3$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.05724

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Abstract of a paper by Luis Garcia-Lirola and Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Unconditional almost squareness and applications to spaces of Lipschitz functions” by Luis Garcia-Lirola and Abraham Rueda Zoca. Abstract: We introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of any dual almost square Banach space. We also take advantage of this notion to provide some criterion of non-duality of some subspaces of scalar as well as vector valued Lipschitz functions. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.04699

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Abstract of a paper by Asuman G. Aksoy and Qidi Peng
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “On a Theorem of S. N. Bernstein for Banach Spaces” by Asuman G. Aksoy and Qidi Peng. Abstract: This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We prove that if $X$ is an infinite-dimensional Banach space and $(Y_n)$ is a nested sequence of subspaces of $X$ such that $Y_n\subset Y_{n+1}$ and $\bar{Y_n}\subset Y_{n+1}$ for any $n\in\mathbb{N}$ and if $(d_n)$ be a decreasing sequence of positive numbers tending to 0, then for any $0<c\leq 1$ there exists $x_c\in X$ such that the distance $\rho (x_c, Y_n)$ from $x_c$ to $Y_n$ satisfies $$cd_n\leq\rho(x_c, Y_n)\leq 4c d_n$$. We prove the above by first improving Borodin's result \cite{Borodin} for Banach spaces by weakening the condition on the sequence $(d_n)$. Lastly, we compare subsequences $(d(\phi_n))$ under different choices of $(\phi_n)$ and examine their effects on approximation. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.04592

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Abstract of a paper by Y. Deng, M. O’Brien, V. G. Troitsky
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Unbounded Norm Convergence in Banach Lattices” by Y. Deng, M. O’Brien, V. G. Troitsky. Abstract: A net $(x_\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x\in X$ if $|x_\alpha-x|\hat u$ converges to 0 in order for all $u\in X_+$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_\alpha)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\||x_\alpha-x|\hat u\|\rightarrow 0$ for all $u\in X_+$. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.03538

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Abstract of a paper by Ryan M. Causey
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Szlenk and $w^*$-dentability indices of $C(K)$” by Ryan M. Causey. Abstract: Given any compact, Hausdorff space $K$ and $1<p<\infty$, we compute the Szlenk and $w^*$-dentability indices of the spaces $C(K)$ and $L_p(C(K))$ We show that if $K$ is compact, Hausdorff, scattered, $CB(K)$ is the Cantor-Bendixson index of $K$, and $\eta$ is the minimum ordinal such that $CB(K)\leq \omega\eta$, then $S_z(C(K))=\omega\eta$ and $D_z(C(K))=S_z(L_p(C(K)))=\omega_1+eta$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.01969

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