Delgado, Pamela
(2020)
Cesàro averaging and extension of functionals on infinite dimensional spaces.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
On the sequence space $\ell^{\infty}$, we construct Banach limits that are invariant under the Ces\`aro averaging operator. On the function space $L^{\infty}(0,\infty)$, we start by defining a new operator $J^{\alpha}$, for each $\alpha >0$. This new operator extends the definition of $J^{n}$, with $n \in \mathbb{N}$, which is the operator obtained by composing the Ces\`aro averaging operator with itself $n$ times. We show that the family of operators $\left(J^{\alpha} \right)_{\alpha >0}$ has the semigroup property. We also construct Banach limits on $L^{\infty}(0,\infty)$ that are invariant under the members of this family of operators. Finally, on the operator space $\mathcal{B}(\ell^2(\mathbb{N}_0))$, we define a Ces\`aro averaging operator from this space to itself. We also discuss known results about vectorvalued Banach limits on $\ell^{\infty}(\ell^2(\mathbb{Z}))$ that preserve Ces\`aro convergence, and use them to construct a continuous linear functional on $\mathcal{B}(\ell^{2}(\mathbb{N}_0))$ with Ces\`aroinvariancelike properties.
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Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 

ETD Committee: 

Date: 
16 September 2020 
Date Type: 
Publication 
Defense Date: 
8 July 2020 
Approval Date: 
16 September 2020 
Submission Date: 
7 September 2020 
Access Restriction: 
2 year  Restrict access to University of Pittsburgh for a period of 2 years. 
Number of Pages: 
112 
Institution: 
University of Pittsburgh 
Schools and Programs: 
Dietrich School of Arts and Sciences > Mathematics 
Degree: 
PhD  Doctor of Philosophy 
Thesis Type: 
Doctoral Dissertation 
Refereed: 
Yes 
Uncontrolled Keywords: 
Functional Analysis, Ces\`aro averaging operators, invariant Banach limits, fractional powers 
Date Deposited: 
16 Sep 2020 13:41 
Last Modified: 
16 Sep 2022 05:15 
URI: 
http://dscholarship.pitt.edu/id/eprint/39721 
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