Holomorphic Functional Calculus - Functional Calculus For A Bounded Operator

Functional Calculus For A Bounded Operator

Let X be a complex Banach space, and L(X) denote the family of bounded operators on X.

Recall the Cauchy integral formula from classical function theory. Let

be holomorphic on some open subset D in the complex plane, and be a rectifiable Jordan curve in D, that is, a closed curve of finite length without self-intersections. Cauchy's integral formula states

for any z lying in the inside of Γ, i.e. the winding number of Γ about z is 1.

The idea is to extend this formula to functions taking values in the Banach space L(X). Cauchy's integral formula suggests the following definition (purely formal, for now):

where (ζ - T)-1 is the resolvent of T at ζ.

Assuming this Banach space-valued integral is appropriately defined, this proposed functional calculus implies the following necessary conditions:

1. As the scalar version of Cauchy's integral formula applies to holomorphic f, we anticipate that is also the case for the Banach space case, where there should be a suitable notion of holomorphy for functions taking values in the Banach space L(X).
2. As the resolvent mapping ζ → (ζ - T)-1 is undefined on the spectrum of T, σ(T), the Jordan curve Γ should not intersect σ(T). Furthermore, the resolvent mapping is holomorphic on the complement of σ(T). So, to obtain a non-trivial functional calculus, Γ must enclose, at least part of, σ(T).
3. The functional calculus should be well-defined in the sense that f(T) is independent of Γ.

The full definition of the functional calculus is as follows: For T ∈ L(X), define

where f is a holomorphic function defined on an open set D in the complex plane which contains σ(T), and

is a collection of Jordan curves in D such that σ(T) lies in the inside of Γ, and each γ_i is oriented in the positive sense.

The open set D may vary with ƒ and need not be connected, as shown by the figures on the right.

The following subsections make precise the notions invoked in the definition and show ƒ(T) is indeed well defined under given assumptions.