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 A_{U} = {T ∈ A;σ(T) ⊂ U} into A such that
f(T) commutes with all operators of finite rank which commute with T. Let
E_{λ}(T) be the projection of the Hilbert space onto ker(λI − T)^{ν(λ)} such that
kerE_{λ}(T) = (λI − T)^{ν(λ)}H. If f defines a strong action T → f(T) in A, then for
each T ∈ A_{U} and λ ∈ σ_{0}(T) there exist a_{0},a_{1},⋯,a_{ν(λ)−1} ∈ C such that
f(T)E_{λ}(T) = ∑
_{j=0}^{ν(λ)−1}(1∕j!)a_{j}(T −λI)^{j}E_{A}(T) (Theorem 3.4). If an algebra ℳ of
functions defines what is called a Daction φ in A, then, in fact, there exists a system
of derivations {D_{k} : 0 ≦ k < m} from ℳ into the algebra of all functions on U such
that φ(f,T)E_{λ}(T) = Σ_{j=0}^{(λ)−1}(1∕j!)D_{j}f(λ)(T −λI)^{j}E_{λ}(T) for all f ∈ℳ,T ∈ A_{7f}
and λ ∈ σ_{0}(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 Daction 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!)D_{j}f(λ)(T − λI)^{j}E_{λ}(T).
