17.1 Compatible Couples
”In the presence of so many different interpolation methods it seemed timely to study the general structure of all possible methods: to determine all of them and to analyze the properties which are common to all.” — [Page 51, AG64].
Definition 17.1.1 (Compatible Couple).label Let $E_{0}, E_{1}, \mathcal{U}$ be topological vector spaces over $K \in \RC$ and $\iota_{0} \in L(E_{0}; \mathcal{U})$ and $\iota_{1} \in L(E_{1}; \mathcal{U})$ be continuous injections. Under the identification that $E_{0}$ and $E_{1}$ are subspaces of $\mathcal{U}$, the pair $(E_{0}, E_{1})$ forms a compatible couple of topological vector spaces.
Remark 17.1.1.label The structure of the compatible couple depends on the common space and the inclusion maps. As such, the couple $(E_{0}, E_{1})$ always implicitly carries the common space and the injections.
Definition 17.1.2 (Sum and Intersection Spaces).label Let $(E_{0}, E_{1})$ be a compatible couple of topological vector spaces over $K \in \RC$, then $E_{0} \cap E_{1}$ is their intersection space, and $E_{0} + E_{1}$ is their sum space.
Definition 17.1.3.label Let $(E_{0}, E_{1})$ be a compatible couple of normed vector spaces over $K \in \RC$, then:
- (1)
$E_{0} \cap E_{1}$ is a normed space under the norm
\[\norm{\cdot}_{E_0 \cap E_1}: E_{0} \cap E_{1} \to [0, \infty) \quad x \mapsto \max(\norm{x}_{E_0}, \norm{x}_{E_1})\] - (2)
$E_{0} + E_{1}$ is a normed space under the norm
\[\norm{\cdot}_{E_0 + E_1}: E_{0} + E_{1} \to [0, \infty)\]with
\[x \mapsto \inf\bracsn{\norm{x_0}_{E_0} + \norm{x_1}_{E_1}|x_0 \in E_0, x_1 \in E_1, x = x_0 + x_1}\] - (3)
If $E_{0}$ and $E_{1}$ are Banach spaces, then $E_{0} \cap E_{1}$ and $E_{0} + E_{1}$ are also Banach spaces.
The norms on $E_{0} \cap E_{1}$ and $E_{0} + E_{1}$ defined above are the intersection and sum norms of the couple, respectively.
Proof. (2): Let $x, y \in E_{0} + E_{1}$, $x_{0}, y_{0} \in E_{0}$, $x_{1}, y_{1} \in E_{1}$ such that $x = x_{0} + x_{1}$ and $y = y_{0} + y_{1}$, then
As this holds for all choices of $x_{0}, y_{0} \in E_{0}$ and $x_{1}, y_{1} \in E_{1}$,
(3): Let $\seq{x_n}\subset E_{0} + E_{1}$ such that $\sum_{n \in \natp}\norm{x_n}_{E_0 + E_1}< \infty$. For each $n \in \natp$, let $y_{n} \in E_{0}$ and $z_{n} \in E_{1}$ with $x_{n} = y_{n} + z_{n}$ and $\norm{y_n}_{E_0}+ \norm{z_n}_{E_1}\le 2\norm{x_n}_{E_0 + E_1}$. Since $E_{0}$ and $E_{1}$ are both complete, $y = \sum_{n = 1}^{\infty} y_{n}$ exists in $E_{0}$ and $z = \sum_{n = 1}^{\infty} z_{n}$ exists in $E_{1}$. Let $x = y + z$, then for each $N \in \natp$,
as $N \to \infty$. Therefore $E_{0} + E_{1}$ is also a Banach space.$\square$
Definition 17.1.4 (Category of Compatible Couples).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$ and $(E_{0}, E_{1})$ be a compatible couple, then $E_{0} E_{1}$ are a compatible couple in $\catc$ if $E_{0}, E_{1}, E_{0} \cap E_{1}, E_{0} + E_{1} \in \catc$.
Let $(E_{0}, E_{1})$ and $(F_{0}, F_{1})$ be compatible couples in $\catc$ and $T \in L(E_{0} + E_{1}, F_{0} + F_{1})$, then $T$ is a morphism of compatible couples if $T|_{E_0}\in \text{Mor}_{\catc}(E_{0}; F_{0})$ and $T|_{E_1}\in \text{Mor}(E_{1}; F_{1})$.
The collection $\catc_{1}$ of all compatible couples in $\catc$ equipped with the above definition of morphisms is the category of compatible couples in $\catc$.
Definition 17.1.5 (Intermediate Space).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $(E_{0}, E_{1}) \in \catc_{1}$ be a compatible couple in $\catc$, and $E \in \catc$, then $E$ is an intermediate space between $E_{0}$ and $E_{1}$ if there exists continuous inclusions
Definition 17.1.6 (Interpolation Spaces).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $(E_{0}, E_{1}), (F_{0}, F_{1}) \in \catc_{1}$ be a compatible couple in $\catc$, and $E, F \in \catc$, then $E$ and $F$ are interpolation spaces with respect to $(E_{0}, E_{1})$ and $(F_{0}, F_{1})$ if
- (1)
$E$ is an intermediate space between $E_{0}$ and $E_{1}$.
- (2)
$F$ is an intermediate space between $F_{0}$ and $F_{1}$.
- (3)
For any $T \in \text{Mor}_{\catc_1}((E_{0}, E_{1}); (F_{0}; F_{1}))$, $T|_{E}\in \text{Mor}_{\catc}(E; F)$.
Definition 17.1.7 (Interpolation Functor).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $\catc_{1}$ be its categories of compatible couples, and $F: \catc_{1} \to \catc$ be a functor, then $F$ is an interpolation functor if for every $(E_{0}, E_{1}), (F_{0}, F_{1}) \in \catc_{1}$,
- (1)
$F((E_{0}, E_{1}))$ and $F((F_{0}, F_{1}))$ are interpolation spaces with respect to $(E_{0}, E_{1})$ and $(F_{0}, F_{1})$.
- (2)
For each $T \in \text{Mor}_{\catc_1}((E_{0}, E_{1}); (F_{0}, F_{1}))$, $F(T) = T|_{F((E_0, E_1))}$.
Definition 17.1.8 (Interpolation Exponent).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $\catc_{1}$ be its categories of compatible couples, $F: \catc_{1} \to \catc$ be an interpolation functor, and $\theta \in [0, 1]$, then $F$ is of exponent $\theta$ if there exists $C \ge 0$ such that for every $(E_{0}, E_{1}), (F_{0}, F_{1}) \in \catc_{1}$ and $T \in \text{Mor}_{\catc_1}((E_{0}, E_{1}); (F_{0}, F_{1}))$
If $C = 1$, then $F$ is of exact exponent $\theta$.
”This is how things appeared in 1965. Fifteen years later, it was found that the number of interpolation methods at our disposal is not large.” — [Page vi, Footnote 3, BK91].
The above quotes are taken from [Page 427, Pie07].
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