Abstract
This note comprises a synthesis of certain results in the theory of exact interpolation between Hilbert spaces. In particular, we examine various characterizations of interpolation spaces and their relations to a number of results in operator theory and in function theory.
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Appendix: The Complex Method is Quadratic
Appendix: The Complex Method is Quadratic
Let \(S=\{z\in \mathbf {C};\, 0\le \operatorname {Re} z\le 1\}\). Fix a Hilbert couple \(\overline {{\mathcal {H}}}\) and let \({\mathcal {F}}\) be the set of functions S → Σ which are bounded and continuous in S, analytic in the interior of S, and which maps the line j + i R into \({\mathcal {H}}_j\) for j = 0, 1. Fix 0 < θ < 1. The norm in the complex interpolation space \(C_\theta \left (\overline {{\mathcal {H}}}\,\right )\) is defined by
Let \({\mathcal {P}}\) denote the set of polynomials \(f=\sum _1^N a_iz^i\) where a i ∈ Δ. We endow \({\mathcal {P}}\) with the inner product
where {P 0, P 1} is the Poisson kernel for S,
Let M θ be the completion of \({\mathcal {P}}\) with this inner product. It is easy to see that the elements of M θ are analytic in the interior of S, and that evaluation map f↦f(θ) is continuous on M θ. Let N θ be the kernel of this functional and define a Hilbert space \({\mathcal {H}}_\theta \) by
We denote the norm in \({\mathcal {H}}_\theta \) by ∥⋅∥θ.
Proposition A.1
\(C_\theta \left (\overline {{\mathcal {H}}}\,\right )={\mathcal {H}}_\theta \) with equality of norms.
Proof
Let \(f\in {\mathcal {F}}\). By the Calderón lemma in [7, Lemma 4.3.2], we have the estimate
Applying Jensen’s inequality, this gives that
Hence \({\mathcal {H}}_\theta \subset C_\theta (\overline {{\mathcal {H}}})\) and \(\left \|\cdot \right \|{ }_{C_\theta (\overline {{\mathcal {H}}})}\le \|\cdot \|{ }_\theta \). On the other hand, for \(f\in {\mathcal {P}}\) one has the estimates
whence \(C_\theta (\overline {{\mathcal {H}}})\subset {\mathcal {H}}_\theta \) and \(\left \|\cdot \right \|{ }_{C_\theta (\overline {{\mathcal {H}}})}\ge \left \|\cdot \right \|{ }_\theta \). □
It is well known that the method C θ is of exponent θ (see, e.g., [7]). We have shown that C θ is an exact quadratic interpolation method of exponent θ.
1.1 Complex Interpolation with Derivatives
In [15, pp. 421–422], Fan considers the more general complex interpolation method C θ(n) for the n:th derivative. This means that in (*), one consider representations \(x=\frac 1 {n!}f^{(n)}(\theta )\) where \(f\in {\mathcal {F}}\); the complex method C θ is thus the special case C θ(0). It is shown in [15] that, for n ≥ 1, the C θ(n)-method is represented, up to equivalence of norms, by the quasi-power function \(h(\lambda )=\lambda ^{\,\theta }/(1+\frac {\theta (1-\theta )}n\left |\,\log \lambda \,\right |)^{\,n}\). The complex method with derivatives was introduced by Schechter [37]; for more details on that method, we refer to the list of references in [15].
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Ameur, Y. (2019). Interpolation Between Hilbert Spaces. In: Aleman, A., Hedenmalm, H., Khavinson, D., Putinar, M. (eds) Analysis of Operators on Function Spaces. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-14640-5_4
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