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# Numerical Ranges and Commutation Properties of Hilbert Space Operators

## 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-1B-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 ΔT is the derivation induced by T ∈ (β(X), then

V(β(β(X)), ΔT) = 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(Tn) ⊂ {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.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
• De Prima, Charles R. (advisor)
Thesis Committee:
• De Prima, Charles R. (chair)
• Taussky-Todd, Olga (co-chair)
Defense Date:31 July 1973
Funders:
Funding AgencyGrant Number
CaltechUNSPECIFIED
Ford FoundationUNSPECIFIED
NSFUNSPECIFIED
US Department of EducationUNSPECIFIED
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 PDF - Final Version See Usage Policy. 25MB