Interpolation Between Hilbert Spaces

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.

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

$$\displaystyle \begin{aligned}\left\|\,x\,\right\|{}_{C_\theta\left(\overline{{\mathcal{H}}}\,\right)}=\inf\left\{\left\|\,f\,\right\|{}_{\mathcal{F}};\, f(\theta)=x\right\}.\end{aligned}$$

(*)

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

$$\displaystyle \begin{aligned}\left\langle f,g\right\rangle_{M_\theta}=\sum_{j=0,1}\int_{\mathbf{R}}\left\langle f(j+it),g(j+it)\right\rangle_jP_j(\theta,t)\, dt,\end{aligned}$$

where {P ₀, P ₁} is the Poisson kernel for S,

$$\displaystyle \begin{aligned}P_j(\theta,t)=\frac {e^{-\pi t}\sin\theta\pi}{\sin^2\theta\pi+(\cos\theta\pi-(-1)^je^{-\pi t})^2}.\end{aligned}$$

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

$$\displaystyle \begin{aligned}{\mathcal{H}}_\theta=M_\theta/N_\theta.\end{aligned}$$

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

$$\displaystyle \begin{aligned}\log\left\|\,f(\theta)\,\right\|{}_{C_\theta(\overline{{\mathcal{H}}})}\le\sum_{j=0,1}\int_{\mathbf{R}}\log\|f(j+it)\|{}_jP_j(\theta,t)\, dt.\end{aligned}$$

Applying Jensen’s inequality, this gives that

$$\displaystyle \begin{aligned}\left\|\,f(\theta)\,\right\|{}_{C_\theta(\overline{{\mathcal{H}}})}\le (\sum_{j=0,1}\int_{\mathbf{R}} \left\|\,f(j+it)\,\right\|{}_j^2P_j(\theta,t)\, dt)^{1/2}=\left\|\,f\,\right\|{}_{M_\theta}.\end{aligned}$$

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

$$\displaystyle \begin{aligned}\left\|\,f(\theta)\,\right\|{}_\theta\le \left\|\,f\,\right\|{}_{M_\theta}\le\sup\{\left\|\,f(j+it)\,\right\|{}_j;\, t\in \mathbf{R},\, j=0,1\}= \left\|\,f\,\right\|{}_{\mathcal{F}},\end{aligned}$$

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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