Vol. 35, No. 1, 1970

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Derivations and actions

Frances F. Gulick

Vol. 35 (1970), No. 1, 95–116

This paper continues the study of functions which act in a Banach algebra containing compact operators on a Hilbert space. In particular, a strong action of a function f in A is considered, that is, an action T f(T) from AU = {T A;σ(T) U} into A such that f(T) commutes with all operators of finite rank which commute with T. Let Eλ(T) be the projection of the Hilbert space onto ker(λI T)ν(λ) such that kerEλ(T) = (λI T)ν(λ)H. If f defines a strong action T f(T) in A, then for each T AU and λ σ0(T) there exist a0,a1,,aν(λ)1 C such that f(T)Eλ(T) = j=0ν(λ)1(1∕j!)aj(T λI)jEA(T) (Theorem 3.4). If an algebra of functions defines what is called a D-action φ in A, then, in fact, there exists a system of derivations {Dk : 0 k < m} from into the algebra of all functions on U such that φ(f,T)Eλ(T) = Σj=0(λ)1(1∕j!)Djf(λ)(T λI)jEλ(T) for all f ∈ℳ,T A7f and λ σ0(T) (Theorem 4.2). Finally, it is shown that if defines an action φ in A, the function x(t) = t is in and φ(x,) is a D-action of x in A which is continuous when restricted to {T A : σ(T) [|z| < r]}, then for every analytic function f ∈ℳφ(f,T) is defined by a sum involving the Cauchy integral formula and terms of the form (1∕j!)Djf(λ)(T λI)jEλ(T).

Mathematical Subject Classification 2000
Primary: 46L20
Secondary: 47A60
Received: 20 February 1969
Revised: 11 February 1970
Published: 1 October 1970
Frances F. Gulick