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(Unpublished)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 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Unpublished |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: |
|
| 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://d-scholarship.pitt.edu/id/eprint/39721 |
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