Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


Dissertationnc.pdf (548.84 KB)

ETD Abstract Container

Abstract Header

On Certain Classes and Ideals of Operators on L1

Riel, Zachariah Charles
http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682

Abstract Details

2016, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
Let (Ω,Σ,μ) be a probability space. An operator T in the space O is an operator on L1(µ) which acts as an operator Tp on Lp(µ) for each 1 ≤ p ≤ ∞. Our aim is to investigate the interaction between the operators Tp for different values of p. For instance, we show that, for an operator T in O, if T1 or T∞ is compact, then Tp is compact for each 1 < p < ∞. We also study the ideal, R, of operators T:L1(µ)→L1(µ) which are representable. In the case when L1(µ) is infinite-dimensional, we show that neither O nor O∩R are ideals in the space of operators on L1(µ). For an operator T in O, we use factorization of operators to give sufficient conditions on T2 and T∞ for representability of T1. We also consider isometries of L1(µ), and ask when an isometry T of L1(µ) acts as an isometry of Lp(µ) for each 1 ≤ p ≤ ∞. We give a partial solution for this problem. Particularly, for a surjective isometry T of L1(µ), we give necessary and sufficient conditions for T to act as an isometry of Lp(µ) for each 1 ≤ p ≤ ∞.
Joseph Diestel (Advisor)
Victor Lomonosov (Committee Member)
Dmitry Ryabogin (Committee Member)
Artem Zvavitch (Committee Member)
Maxim Dzero (Committee Member)
Austin Melton (Committee Member)
86 p.
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

Recommended Citations

Citations

  • Riel, Z. C. (2016). On Certain Classes and Ideals of Operators on L1 [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682

    APA Style (7th edition)

  • Riel, Zachariah. On Certain Classes and Ideals of Operators on L1. 2016. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682.

    MLA Style (8th edition)

  • Riel, Zachariah. "On Certain Classes and Ideals of Operators on L1." Doctoral dissertation, Kent State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=kent1479753036578682

    Chicago Manual of Style (17th edition)

kent1479753036578682
643
© 2016, all rights reserved.
This open access ETD is published by Kent State University and OhioLINK.