page_title: Holomorphic functional calculus
section_title: Functional calculus for a bounded operator
hierarchical_section_title: Holomorphic functional calculus / Functional calculus for a bounded operator
context_page_description: In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f, which naturally extends the function f from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of T to the bounded operators.
  In particular, T can be a square matrix with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction.
context_section_description: 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 f : C → C be holomorphic on some open set D ⊂ C, and Γ be a rectifiable Jordan curve in D, that is, a closed curve of finite length without self-intersections. Assume that the set U of points lying in the inside of Γ, i.e. such that the winding number of Γ about z is 1, is contained in D. The Cauchy integral formula states
  for any z in U.
  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:
  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).
  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). Now, the resolvent mapping will be holomorphic on the complement of σ(T). So to obtain a non-trivial functional calculus, Γ must enclose (at least part of) σ(T).
  The functional calculus should be well-defined in the sense that f(T) has to be 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 ⊂ C which contains σ(T), and Γ = {γ₁, ..., γₘ} is a collection of disjoint Jordan curves in D bounding an "inside" set U, such that σ(T) lies in U, and each γᵢ is oriented in the boundary sense.
  The open set D may vary with f and need not be connected or simply connected, as shown by the figures on the right.
  The following subsections make precise the notions invoked in the definition and show f(T) is indeed well defined under given assumptions.
caption: The case when the spectrum has multiple connected components and the corresponding path γ.
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