Infinite-dimensional Holomorphy - Vector-valued Holomorphic Functions Defined in The Complex Plane

Vector-valued Holomorphic Functions Defined in The Complex Plane

A first step in extending the theory of holomorphic functions beyond one complex dimension is considering so-called vector-valued holomorphic functions, which are still defined in the complex plane C, but take values in a Banach space. Such functions are important, for example, in constructing the holomorphic functional calculus for bounded linear operators.

A function f:U→X defined on an open subset U of the complex plane with values in a complex Banach space X is called holomorphic if it is complex-differentiable; that is, for each point z in U the limit

exists.

One may define the line integral of a vector-valued holomorphic function f:U→X along a rectifiable curve γ → U in the same way as for complex-valued holomorphic functions, as the limit of sums of the form

where a = t₀ < t₁ < ... < t_n = b is a subdivision of the interval, as the lengths of the subdivision intervals approach zero.

It is a quick check that the Cauchy integral theorem also holds for vector-valued holomorphic functions. Indeed, if f:U→X is such a function and T:X→C a bounded linear functional, one can show that

Moreover, the composition Tof:U→C is a complex-valued holomorphic function. Therefore, for γ a simple closed curve whose interior is contained in U, the integral on the right is zero, by the classical Cauchy integral theorem. Then, since T is arbitrary, it follows from the Hahn–Banach theorem that

which proves the Cauchy integral theorem in the vector-valued case.

Using this powerful tool one may then prove Cauchy's integral formula, and, just like in the classical case, that any vector-valued holomorphic function is analytic.

A useful criterion for a function f : U → X to be holomorphic is that T o f : U → C is a holomorphic complex-valued function for every continuous linear functional T : X → C. Such an f is weakly holomorphic. It can be shown that a function defined on an open subset of the complex plane with values in a Fréchet space is holomorphic if, and only if, it is weakly holomorphic.