On Certain Classes and Ideals of Operators on L<sub>1</sub>
| Main Author: | |
|---|---|
| Language: | English |
| Published: |
Kent State University / OhioLINK
2016
|
| Subjects: | |
| Online Access: | http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682 |
| id |
ndltd-OhioLink-oai-etd.ohiolink.edu-kent1479753036578682 |
|---|---|
| record_format |
oai_dc |
| spelling |
ndltd-OhioLink-oai-etd.ohiolink.edu-kent14797530365786822021-08-03T06:38:56Z On Certain Classes and Ideals of Operators on L<sub>1</sub> Riel, Zachariah Charles Mathematics Functional analysis operators on Lebesgue spaces vector measures compact operators completely continuous operators strictly singular operators nuclear operators absolutely summing operators representable operators isometries Let (Ω,Σ,μ) be a probability space. An operator T in the space <i>O</I> is an operator on L<sub>1</sub>(µ) which acts as an operator T<sub>p</sub> on L<sub>p</sub>(µ) for each 1 ≤ p ≤ ∞. Our aim is to investigate the interaction between the operators T<sub>p</sub> for different values of p. For instance, we show that, for an operator T in <i>O</I>, if T<sub>1</sub> or T<sub>∞</sub> is compact, then T<sub>p</sub> is compact for each 1 < p < ∞. We also study the ideal, <i>R</i>, of operators T:L<sub>1</sub>(µ)→L<sub>1</sub>(µ) which are representable. In the case when L<sub>1</sub>(µ) is infinite-dimensional, we show that neither <i>O</i> nor <i>O</i>∩<I>R</i> are ideals in the space of operators on L<sub>1</sub>(µ). For an operator T in <i>O</i>, we use factorization of operators to give sufficient conditions on T<sub>2</sub> and T<sub>∞</sub> for representability of T<sub>1</sub>. We also consider isometries of L<sub>1</sub>(µ), and ask when an isometry T of L<sub>1</sub>(µ) acts as an isometry of L<sub>p</sub>(µ) for each 1 ≤ p ≤ ∞. We give a partial solution for this problem. Particularly, for a surjective isometry T of L<sub>1</sub>(µ), we give necessary and sufficient conditions for T to act as an isometry of L<sub>p</sub>(µ) for each 1 ≤ p ≤ ∞. 2016-11-22 English text Kent State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682 http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
| collection |
NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics Functional analysis operators on Lebesgue spaces vector measures compact operators completely continuous operators strictly singular operators nuclear operators absolutely summing operators representable operators isometries |
| spellingShingle |
Mathematics Functional analysis operators on Lebesgue spaces vector measures compact operators completely continuous operators strictly singular operators nuclear operators absolutely summing operators representable operators isometries Riel, Zachariah Charles On Certain Classes and Ideals of Operators on L<sub>1</sub> |
| author |
Riel, Zachariah Charles |
| author_facet |
Riel, Zachariah Charles |
| author_sort |
Riel, Zachariah Charles |
| title |
On Certain Classes and Ideals of Operators on L<sub>1</sub> |
| title_short |
On Certain Classes and Ideals of Operators on L<sub>1</sub> |
| title_full |
On Certain Classes and Ideals of Operators on L<sub>1</sub> |
| title_fullStr |
On Certain Classes and Ideals of Operators on L<sub>1</sub> |
| title_full_unstemmed |
On Certain Classes and Ideals of Operators on L<sub>1</sub> |
| title_sort |
on certain classes and ideals of operators on l<sub>1</sub> |
| publisher |
Kent State University / OhioLINK |
| publishDate |
2016 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682 |
| work_keys_str_mv |
AT rielzachariahcharles oncertainclassesandidealsofoperatorsonlsub1sub |
| _version_ |
1719440807150223360 |