# Cesàro averaging and extension of functionals on infinite dimensional spaces

Delgado, Pamela (2020) Cesàro averaging and extension of functionals on infinite dimensional spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## 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 vector-valued 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\aro-invariance-like properties.

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## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
ETD Committee:
Committee ChairLennard, Christopherlennard@pitt.edu
Committee MemberHajlasz, Piotrhajlasz@pitt.edu
Committee MemberGartside, Paulpmg20@pitt.edu
Committee MemberTurett, Barryturett@oakland.edu
Committee MemberDowling, Patrickdowlinpn@miamioh.edu
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