21 research outputs found
Abstract Ces\`aro spaces: Integral representations
The Ces\`aro function spaces , , have
received renewed attention in recent years. Many properties of are
known. Less is known about when the Ces\`aro operator takes its values
in a rearrangement invariant (r.i.) space other than . In this paper
we study the spaces via the methods of vector measures and vector
integration. These techniques allow us to identify the absolutely continuous
part of and the Fatou completion of ; to show that is
never reflexive and never r.i.; to identify when 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
; it is never compact but, it can be completely continuous.Comment: 21 page
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 with non-trivial Boyd
indices, thereby extending Widom's results for 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
Giovanni Battista Guccia: pioneer of international cooperation in mathematics
This book examines the life and work of mathematician Giovanni Battista Guccia, founder of the Circolo Matematico di Palermo and its renowned journal, the Rendiconti del Circolo matematico di Palermo. The authors describe how Guccia, an Italian geometer, was able to establish a mathematical society in Sicily in the late nineteenth century, which by 1914 would grow to become the largest and most international in the world, with one of the most influential journals of the time. The book highlights the challenges faced by Guccia in creating an international society in isolated Palermo, and places Guccia’s activities in the wider European context through comparisons with the formation of the London Mathematical Society and the creation of Mittag-Leffler’s Acta Mathematica in Stockholm. Based on extensive searches in European archives, this scholarly work follows both historical and scientific treads, and will appeal to those interested in the history of mathematics and science in general
Extensions of the classical Cesaro operator on Hardy spaces
For each 1\le p<\infty, the classical Cesà ro operator from the Hardy space to itself has the property that there exist analytic functions with . This article deals with the identification and properties of the (Banach) space consisting of all analytic functions that maps into . It is shown that contains classical Banach spaces of analytic functions , genuinely bigger that , such that has a continuous -valued extension to . An important feature is that is the largest amongst all such spaces