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
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- 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 Xj to Yj, 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.
Read more about this topic: Interpolation Space
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