Θα θέλαμε να σας ενημερώσουμε ότι την Τρίτη 18 Ιουνίου στην αίθουσα Μ2, 3ος Όροφος της Σ.Θ.Ε. και ώρα 11:15-12:00, στο πλαίσιο του Σεμιναρίου του Τομέα Μαθηματικής Ανάλυσης, θα δώσει ομιλία ο Alejandro Mahillo (Universidad de Zaragoza).
Τίτλος: Spectral properties of the Cesàro operator via \(C_0\)-semigroup theory and a note on universality
Περίληψη: In 1965, Brown, Halmos, and Shields investigated the boundedness of the operators \(C_0\), \(C_1\), and \(C_\infty\), defined on the spaces \(\ell^2\), \(L^2[0,1]\), and \(L^2(0,\infty)\) respectively, as follows:
\[ C_0 f(n) = \frac{1}{n+1}\sum_{k=0}^n f(k), \quad C_1 f(t) = \frac{1}{t}\int_0^t f(s)\,ds, \quad C_\infty f(t) = \frac{1}{t}\int_0^t f(s)\,ds. \]
This foundational work was later extended to the range \(1 < p \leq \infty\), where the spectral properties of these operators were analyzed. Specifically, in 1968, Boyd explored the operator \(C_\infty\) on \(L^p(0,\infty)\), and during the early 1970s, Leibowitz studied \(C_0\) and \(C_1\) on the spaces \(\ell^p\) and \(L^p[0,1]\), respectively.
In more recent developments, detailed analyses were conducted in 2014 and 2018 on the boundedness and spectral characteristics of \(C_\infty\) and \(C_0\), along with some generalized Cesàro operators, on \(L^p(0,\infty)\) and \(\ell^p\) through the lens of strongly continuous semigroups.
In this presentation, we will delve into the generalized Cesàro operator \(C_\alpha\) for \(\alpha > 0\), defined on \(L^p[0,1]\) by:
\[C_\alpha f(t) = \frac{\alpha}{t^\alpha} \int_0^t (t-s)^{\alpha-1} f(s)\,ds.\]
Our focus will be on representing this operator through strongly continuous semigroups of composition operators, allowing us to uncover intricate properties of the spectrum of \(C_\alpha\). Moreover, we will investigate the universality of the associated composition semigroup on \(L^2[0,1]\).