Interpolation Space - Complex Interpolation

Complex Interpolation

If the scalars are complex numbers, properties of complex analytic functions are used to define an interpolation space. Given a compatible couple (X₀, X₁) of Banach spaces, the linear space consists of all analytic functions f with values in X₀+X₁, defined on the open strip S = { z : 0 < Re z < 1 } in the complex plane, continuous on the closed strip 0 ≤ Re z ≤ 1, such that the set of values f(z), z ∈ S, is bounded in X₀ + X₁ and such that

the set of values { f (i t ) : t ∈ R } is bounded in X₀ and { f (1+ it) : t ∈ R } is bounded in X₁.

This space of functions is a Banach space under the norm

Definition. For 0 < θ < 1, the complex interpolation space (X₀, X₁)_θ is the linear subspace of X₀ + X₁ consisting of all values f(θ) when f varies in the preceding space of functions,

The norm on the complex interpolation space (X₀, X₁)_θ is defined by

Equipped with this norm, the complex interpolation space (X₀, X₁)_θ is a Banach space.

Theorem. Given two compatible couples of Banach spaces (X₀, X₁) and (Y₀, Y₁), the pair ((X₀, X₁)_θ, (Y₀, Y₁)_θ) is an exact interpolation pair of exponent θ, i.e., if T is a linear operator from X₀ + X₁ to Y₀ + Y₁, bounded from X_j to Y_j, j = 0, 1, then T is bounded from (X₀, X₁)_θ to (Y₀, Y₁)_θ and

The family of Lp spaces (consisting of complex valued functions) behaves well under complex interpolation. If (R, Σ, μ) is an arbitrary measure space, if 1 ≤ p₀, p₁ ≤ ∞ and 0 < θ < 1, then

with equality of norms. This fact is closely related to the Riesz–Thorin theorem.