Abstract Ces\`aro spaces: Integral representations

The Ces\`aro function spaces $Ces_p=[C,L^p]$, $1\le p\le\infty$, have received renewed attention in recent years. Many properties of $[C,L^p]$ are known. Less is known about $[C,X]$ when the Ces\`aro operator takes its values in a rearrangement invariant (r.i.) space $X$ other than $L^p$. In this paper we study the spaces $[C,X]$ via the methods of vector measures and vector integration. These techniques allow us to identify the absolutely continuous part of $[C,X]$ and the Fatou completion of $[C,X]$; to show that $[C,X]$ is never reflexive and never r.i.; to identify when $[C,X]$ is weakly sequentially complete, when it is isomorphic to an AL-space, and when it has the Dunford-Pettis property. The same techniques are used to analyze the operator $C:[C,X]\to X$; it is never compact but, it can be completely continuous.Comment: 21 page

Automobility and Racial Injustice in Spatial Production

From the Washington University Senior Honors Thesis Abstracts (WUSHTA), 2017. Published by the Office of Undergraduate Research. Joy Zalis Kiefer, Director of Undergraduate Research and Associate Dean in the College of Arts & Sciences; Lindsey Paunovich, Editor; Helen Human, Programs Manager and Assistant Dean in the College of Arts and Sciences Mentor: Carol Camp Yeake

The Cesàro operator on weighted l_1 spaces

This is the peer reviewed version of the following article: Albanese, Angela, Bonet Solves, José Antonio, Ricker, W.. (2018). The Cesàro operator on weighted l_1 spaces.Mathematische Nachrichten, 291, 7, 1015-1048, which has been published in final form at http://doi.org/10.1002/mana.201600509. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] Unlike for l(p), 1 < p <= infinity, the discrete Cesaro operator C does not map l(1) into itself. We identify precisely those weights w such that C does map l(1)(w) continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of C are presented. It is also possible to identify all w such that C is a compact operator in l(1)(w). The final section investigates the mean ergodic properties of C in l(1)(w). Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.Ministry of Education, Science and Art, Bavaria (Germany); Grant: International Visiting Professor Program 2016; Ministry of Economy of Spain; Grant: projects MTM2013-43540-P and MTM2016-76647-PAlbanese, A.; Bonet Solves, JA.; Ricker, W. (2018). The Cesàro operator on weighted l_1 spaces. Mathematische Nachrichten. 291(7):1015-1048. https://doi.org/10.1002/mana.201600509S10151048291

Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces

We consider the problem of extending or factorizing a bounded bilinear map defined on a couple of order continuous Banach function spaces to its optimal domain, i.e. the biggest couple of Banach function spaces to which the bilinear map can be extended. As in the case of linear operators, we use vector measure techniques to find this space, and we show that this procedure cannot be always successfully used for bilinear maps. We also present some applications to find optimal factorizations of linear operators between Banach function spaces.J. M. Calabuig was supported by Ministerio de Economia y Competitividad (Spain) (project MTM2011-23164) and by "Jose Castillejo 2009" (MEC). E. A. Sanchez-Perez was supported by MEC and FEDER (project MTM2009-14483-C02-02). J. M. Calabuig and E. A. Sanchez-Perez were also supported by Ayuda para Estancias de PDI de la UPV en Centros de Investigacion de Prestigio (PAID-00-11).Calabuig Rodriguez, JM.; Fernandez Unzueta, M.; Galaz Fontes, F.; Sánchez Pérez, EA. (2014). Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas (RACSAM). 108(2):353-367. https://doi.org/10.1007/s13398-012-0101-7S3533671082Calabuig, J.M., Galaz-Fontes, F., Jiménez Fernández, E., Sánchez Pérez, E.A.: Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series. Math. Z. 257, 381–402 (2007)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Curbera, G.P.: Operators into $$L^1$$ of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)Curbera, G.P. , Ricker, W.J.: Optimal domains for the kernel operator associated with Sobolev’s inequality. Studia Math. 158(2), 131–152 (2003) [see also Corrigenda in the same journal, 170 (2005) 217–218)]Delgado, O.: Banach function subspaces of $$L^1$$ of a vector measure and related Orlicz spaces. Indag. Math. (N. S.) 15, 485–495 (2004)Delgado, O.: Optimal domains for kernel operators on $$[0,\infty )\times [0,\infty )$$ . Studia Math. 174, 131–145 (2006)Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007)Diestel, J., Uhl, J.J.: Vector measures. In: Math. Surveys, vol. 15. Amer. Math. Soc., Providence (1977)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Galdames, O., Sánchez Pérez, E.A.: Optimal range theorems for operators with $$p$$ -th power factorable adjoints. Banach J. Math. Anal. 6(1), 61–73 (2012)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. In: Oper. Theory Adv. Math. Appl., vol. 180. Birkäuser, Basel (2008

Fine spectra of the finite Hilbert transform in function spaces

We investigate the spectrum and fine spectra of the finite Hilbert transform acting on rearrangement invariant spaces over $(-1,1)$ with non-trivial Boyd indices, thereby extending Widom's results for $L^p$ spaces. In the case when these indices coincide, a full description of the spectrum and fine spectra is given.Comment: 26 pages, 1 figure. Minor changes from previous version. This is the final version, to be published in Advances in Mathematic

Extension and Integral Representation of the finite Hilbert Transform In Rearrangement Invariant Spaces

The finite Hilbert transform $T$ is a classical (singular) kernel operator which is continuous in every rearrangement invariant space $X$ over $(-1,1)$ having non-trivial Boyd indices. For $X=L^p$, $1<p<\infty$, this operator has been intensively investigated since the 1940's (also under the guise of the ``airfoil equation''). Recently, the extension and inversion of $T\colon X\to X$ for more general $X$ has been studied in G. P. Curbera, S. Okada, W. J. Ricker, Inversion and extension of the finite Hilbert transform on $(-1,1)$, Ann. Mat. Pura Appl. 198 (2019), 1835-1860, where it is shown that there exists a larger space $[T,X]$, optimal in a well defined sense, which contains $X$ continuously and such that $T$ can be extended to a continuous linear operator $T\colon[T,X]\to X$. The purpose of this paper is to continue this investigation of $T$ via a consideration of the $X$-valued vector measure $m_X\colon A\mapsto T(\chi_A)$ induced by $T$ and its associated integration operator $f\mapsto \int_{-1}^1f\,dm_X$. In particular, we present integral representations of $T\colon X\to X$ based on the $L^1$-space of $m_X$ and other related spaces of integrable functions