On Banach algebra elements of thin numerical range

Abstract

Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W(h) ⊂ , where W(h) denotes the numerical range of h (7), or if ║eiλh║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the conditionis satisfied. These may be termed the elements of thin numerical range if δ is small.