Holomorphic functional calculus

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.

From above it is evident that all that is really needed is the radius of convergence of the MacLaurin series be greater than ǁTǁ, the operator norm of T. This enlarges somewhat the family of f for which f(T) can be defined using the above approach.

For instance, it is a fact from matrix theory that every non-singular T has a logarithm S in the sense that eS = T. It is desirable to have a functional calculus that allows one to define, for a non-singular T, ln(T) such that it coincides with S. This can not be done via power series, for example the logarithmic series converges only on the open unit disk.

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

One can parametrize each γi ∈ Γ by a real interval [a, b], and the integral is the limit of the Riemann sums obtained from ever-finer partitions of [a, b].

We define In the definition of the functional calculus, f is assumed to be holomorphic in an open neighborhood of Γ.

Letting z2 → z1 shows the resolvent map is (complex-) differentiable at each z1 ∈ ρ(T); so the integral in the expression of functional calculus converges in L(X).

To verify this claim, let z1 ∈ ρ(T) and notice the formal expression suggests we consider for (z2−T)−1.

The formal expression leads one to consider This series, the Neumann series, converges to (z−T)−1 if From the last two properties of the resolvent we can deduce that the spectrum σ(T) of a bounded operator T is a compact subset of C. Therefore, for any open set D such that σ(T) ⊂ D, there exists a positively oriented and smooth system of Jordan curves Γ = {γ1, ..., γm} such that σ(T) is in the inside of Γ and the complement of D is contained in the outside of Γ.

Hence, for the definition of the functional calculus, indeed a suitable family of Jordan curves can be found for each f that is holomorphic on some D. The previous discussion has shown that the integral makes sense, i.e. a suitable collection Γ of Jordan curves does exist for each f and the integral does converge in the appropriate sense.

Suppose Γ = {γi} and Ω = {ωj} are two (finite) collections of Jordan curves satisfying the assumption given for the functional calculus.

We wish to show Let Ω′ be obtained from Ω by reversing the orientation of each ωj, then Consider the union of the two collections Γ ∪ Ω′.

For well-definedness, we only needed f to be holomorphic on an open set U containing the contours Γ ∪ Ω′ but not necessarily σ(T).

To prove this, it is sufficient to show, for k ≥ 0 and f(z) = zk, it is true that f(T) = Tk, i.e. for any suitable Γ enclosing σ(T).

Start by calculating directly The last line follows from the fact that ω ∈ Γ2 lies outside of Γ1 and f1 is holomorphic on some open neighborhood of σ(T) and therefore the second term vanishes.

We note that, everything discussed so far holds verbatim if the family of bounded operators L(X) is replaced by a Banach algebra A.

Define e(z) = 1 if z ∈ U and e(z) = 0 if z ∈ V. Then e is a holomorphic function with [e(z)]2 = e(z) and so, for a suitable contour Γ which lies in U ∪ V and which encloses σ(T), the linear operator will be a bounded projection that commutes with T and provides a great deal of useful information.

Any isolated point of σ(T) is both open and closed in the subspace topology and therefore has an associated spectral projection.

Let σ(T) be a disjoint union Define ei to be 1 on some neighborhood that contains only the component Fi and 0 elsewhere.

The mapping R: X' → X defined by is a Banach space isomorphism, and we see that This can be viewed as a block diagonalization of T. When X is finite-dimensional, σ(T) = {λi} is a finite set of points in the complex plane.

The corresponding block-diagonal matrix is the Jordan canonical form of T. With stronger assumptions, when T is a normal operator acting on a Hilbert space, the domain of the functional calculus can be broadened.

When comparing the two results, a rough analogy can be made with the relationship between the spectral theorem for normal matrices and the Jordan canonical form.

In that context, if E ⊂ σ(T) is a Borel set and 1E is the characteristic function of E, the projection operator 1E(T) is a refinement of ei(T) discussed above.

In slightly more abstract language, the holomorphic functional calculus can be extended to any element of a Banach algebra, using essentially the same arguments as above.

A holomorphic functional calculus can be defined in a similar fashion for unbounded closed operators with non-empty resolvent set.