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 ℳ 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 ℳ 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 ℳ defines an action φ in A,
the function x(t) = t is in ℳ 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).
