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 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(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.

Thesis Files