## Citation

Richard, Bruce Kent
(1974)
*Numerical Ranges and Commutation Properties of Hilbert Space Operators.*
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/0j1k-nr94.
https://resolver.caltech.edu/CaltechTHESIS:02042021-163138095

## Abstract

Application of the theory of numerical ranges to the study of commutation properties of operators is the purpose of the thesis.

For a complex, unital Banach algebra ℝ, T ∈ ℝ, the numerical range of T is V(ℝ, T) = {f(T) :f(1) = 1 = ∥f∥, f ∈ ℝ*}. This is a generalization and extension of the notion of the numerical range defined for a bounded operator T on the Hilbert space **H**: W(T) = {(Tx, x):x ∈ **H**, (x,x) = 1}. These numerical range concepts are used in studies of multiplicative commutators, derivations, and powers of accretive operators.

An extension of Frobenius' group commutator theorem is obtained: For T,A,B ∈ β(**H**), T = ABA^{-1}B^{-1}, AT = TA, A normal and 0 ∉W(B)^{-} imply T = 1. Other extensions of the Frobenius theorem are proved and a special discussion is given about these results in the case **H** is finite dimensional. The sharpness of the results is also reviewed.

For X a Banach space, the numerical range of a derivation acting on β(X) is completely characterized. If Δ_{T} is the derivation induced by T ∈ (β(X), then

V(β(β(X)), Δ_{T}) = V(β(X),T) - V(β (X),T).

Normal elements of general Banach algebras are discussed. A consequence of an examination of derivations which are normal is a simple proof of the Fuglede- Putnam Theorem.

A theorem for matrices by C. R. Johnson is generalized to the operator case: for T ∈ (β(**H**), W(T^{n}) ⊂ {Rez ≥ 0}, n = 1, 2,... if and only if T ≥ 0. Examples are given which show neither the necessity nor the sufficiency part of the theorem can be transplanted into the general Banach algebra setting. A containment result for the numerical range of a product is also proved.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||||||
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Subject Keywords: | Mathematics | ||||||||||

Degree Grantor: | California Institute of Technology | ||||||||||

Division: | Physics, Mathematics and Astronomy | ||||||||||

Major Option: | Mathematics | ||||||||||

Thesis Availability: | Public (worldwide access) | ||||||||||

Research Advisor(s): | - De Prima, Charles R. (advisor)
- Taussky-Todd, Olga (co-advisor)
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Thesis Committee: | - De Prima, Charles R. (chair)
- Taussky-Todd, Olga (co-chair)
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Defense Date: | 31 July 1973 | ||||||||||

Funders: |
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Record Number: | CaltechTHESIS:02042021-163138095 | ||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:02042021-163138095 | ||||||||||

DOI: | 10.7907/0j1k-nr94 | ||||||||||

Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||

ID Code: | 14076 | ||||||||||

Collection: | CaltechTHESIS | ||||||||||

Deposited By: | Benjamin Perez | ||||||||||

Deposited On: | 06 Jul 2021 17:25 | ||||||||||

Last Modified: | 13 Jul 2021 22:17 |

## Thesis Files

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