Interpolation of Holomorphic functions

TL;DR

This paper explores conditions under which holomorphic functions maintain their compactness and holomorphic properties when restricted to interpolated Banach spaces, advancing interpolation theory for complex analysis.

Contribution

It provides new criteria ensuring that restrictions of holomorphic maps to interpolated spaces are both compact and holomorphic, extending existing interpolation results.

Findings

Identifies conditions for compactness preservation in holomorphic interpolations

Establishes when holomorphicity is retained after interpolation

Enhances understanding of interpolation in complex Banach spaces

Abstract

Interpolation Theory gives techniques for constructing spaces from two initial Banach spaces. We provide several conditions under which the restriction of a holomorphic map $f:X_0+X_1 \rightarrow Y_0+Y_1$ to the interpolated spaces (using some specific interpolation methods), where $f_{|X_0}:X_0 \rightarrow Y_0$ is compact, is also compact and holomorphic.

Equations

$$\Vert x{\Vert }_{X_0\cap X_1}=\max \{\Vert x{\Vert }_{X_0},\Vert x{\Vert }_{X_1}\},$$

$$\Vert x{\Vert }_{X_0+X_1}=\inf \{\Vert x_0{\Vert }_{X_0}+\Vert x_1{\Vert }_{X_1}:x=x_0+x_1,x_i\in X_i\},$$

$$X_0\cap X_1\subseteq F(X_0,X_1)\subseteq X_0+X_1,$$

$$\hat{f}(k)=\frac{1}{2\pi i}{\int }_0^{2\pi }{e}^{-ikt}f(e^{it})dt.$$

$$\begin{array}[]{crcl}\mathfrak{F}:&L^{2}(\mathbb{T},X)&\longrightarrow&\ell^{2}(X)\\ &f&\mapsto&\{\hat{f}(k)\}_{k=1}^{\infty},\end{array}$$

$$f(y)=\sum _{m=0}^{\infty }{P}_{m}f(x)(y-x),$$

$$P_{m}f(x)=\frac{d^{m}f(x)}{m!}.$$

$$R^{c}(x)=\sup\bigg{\{}R>0\,:\,f(y)=\sum_{m=1}^{\infty}{d^{m}f(x)\over m!}(y-x)\text{ uniformly for }\|y-x\|_{X}\leq r<R\bigg{\}}.$$

$$\tilde{P}(\stackrel{m}{\overbrace{x_1,\dots ,x_m}})=\frac{1}{2^{m}m!}\sum _{{\epsilon }_1,\dots ,{\epsilon }_m=\pm 1}{\epsilon }_1\cdot \dots \cdot {\epsilon }_{m}P({\epsilon }_1{x}_1+\dots +{\epsilon }_m{x}_m).$$

$$\Vert \tilde{P}{\Vert }_{X\to Y}\le \frac{m^m}{m!}\Vert P{\Vert }_{X\to Y}.$$

$$T:(X_0^{(1)}+X_0^{(2)})\times (X_1^{(1)}+X_1^{(2)})\to Y_0\cap Y_1$$

$$\displaystyle\mathcal{F}\{X_{0},X_{1}\}=\big{\{}\varphi:\overline{\mathbb{L}}\rightarrow X_{0}+X_{1}\text{ so that }$$

$$\displaystyle\varphi:\{|z|=e^{j}\}\rightarrow X_{j}\text{ continuously},\,j=0,1\big{\}}.$$

$$\Vert \phi {\Vert }_{\mathcal{F}\{X_0,X_1\}}=\max \left\{\max \{\Vert \phi (e^{it}){\Vert }_{X_0},\Vert \phi (e^{1+it}){\Vert }_{X_1}\}:t\in [0,2\pi ]\right\}.$$

$$[X_0,X_1{]}_{\theta }=\{\phi (e^{\theta }):\phi \in \mathcal{F}\{X_0,X_1\}\},$$

$$\Vert x{\Vert }_{[X_0,X_1{]}_{\theta }}=\inf \{\Vert \phi {\Vert }_{\mathcal{F}\{X_0,X_1\}}:x=\phi (e^{\theta })\text{ for some }\phi \in \mathcal{F}\{X_0,X_1\}\}.$$

$${⟨X_0,X_1⟩}_{\theta }=\{x=\sum _{k\in \mathbb{Z}}{x}_k:$$

$$\text{ and }\sum _{k\in \mathbb{Z}}{e}^{(j-\theta )k}{x}_k\text{ is unconditionally convergent in }{X}_j,j=0,1\}.$$

$$\|x\|_{\left<X_{0},X_{1}\right>_{\theta}}=\inf\max_{j=0,1}\sup\bigg{\|}\sum_{k\in\mathbb{Z}}\lambda_{k}e^{(j-\theta)k}x_{k}\bigg{\|}_{X_{j}},$$

$$\frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1},\frac{1}{q}=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}.$$

$$T:L_{p_0}(U,d\mu )\to L_{q_0}(V,d\nu )\text{and}T:L_{p_1}(U,d\mu )\to L_{q_1}(V,d\nu )$$

$$T:L_p(U,d\mu )\to L_q(V,d\nu )$$

$$T:(X_0^{(1)}\cap X_1^{(1)})\times \cdots \times (X_0^{(n)}\cap X_1^{(n)})\to Y_0\cap Y_1$$

$$T:E:=[X_0^{(1)},X_1^{(1)}{]}_{\theta }\times \cdots \times [X_0^{(n)},X_1^{(n)}{]}_{\theta }\to [Y_0,Y_1{]}_{\theta },$$

$$[X_{0},X_{1}]_{\theta}=\Big{[}\overline{X_{0}\cap X_{1}}^{\|\cdot\|_{0}},X_{1}\Big{]}_{\theta}=\Big{[}X_{0},\overline{X_{0}\cap X_{1}}^{\|\cdot\|_{1}}\Big{]}_{\theta}=\Big{[}\overline{X_{0}\cap X_{1}}^{\|\cdot\|_{0}},\overline{X_{0}\cap X_{1}}^{\|\cdot\|_{1}}\Big{]}_{\theta}.$$

$$[X_{0},X_{1}]_{\theta}=\Big{[}[X_{0},X_{1}]_{0},[X_{0},X_{1}]_{1}\Big{]}_{\theta}.$$

$$\Vert x{\Vert }_{[X_0,X_1{]}_{\theta }}\le C\Vert x{\Vert }_{X_0}^{1-\theta }\Vert x{\Vert }_{X_1}^{\theta }.$$

$$x$$

$$\Vert x_0{\Vert }_{X_0}$$

$$\Vert x_1{\Vert }_{X_1}$$

$$\{\varphi(e^{it})\,:\,t\in[0,2\pi]\}\subseteq\bigcup_{k=1}^{N}B\Big{(}\varphi(e^{it_{k}});{\delta_{t_{k}}^{0}\over 2}\Big{)}.$$

$$\Vert f(\zeta )-f\circ \phi (e^{it}){\Vert }_{Y_0}\le \Vert f(\zeta )-f\circ \phi (e^{i{t}_k}){\Vert }_{Y_0}+\Vert f\circ \phi (e^{i{t}_k})-f\circ \phi (e^{it}){\Vert }_{Y_0}<\epsilon .$$

Full text

1. Introduction and preliminaries

Interpolation theory has proved to be a very important area of study within Functional Analysis and has provided a rich variety of new techniques when studying Banach spaces in general, and $L_{p}$ spaces in particular. We refer the interested reader to [4] and the references therein for a complete introduction to this theory. Interpolation theory is still a very fruitful field of research, and one can consult the references [1, 3, 6, 7, 11, 19, 22, 26, 30] for a sample of recent papers published in this area.

Let $\mathcal{C}$ be the class of all compatible couples of Banach spaces (that is, those pairs $(X_{0},X_{1})$ of Banach spaces for which there exists a topological vector space $\mathcal{A}$ so that $X_{0}\subseteq\mathcal{A}\supseteq X_{1}$ as subspaces). For $(X_{0},X_{1})\in\mathcal{C}$, we can endow $X_{0}\cap X_{1}$ with the norm

[TABLE]

and $(X_{0}\cap X_{1},\|\cdot\|_{X_{0}\cap X_{1}})$ is a Banach space.

Similarly, we may define a norm on $X_{0}+X_{1}$ by

[TABLE]

and $(X_{0}+X_{1},\|\cdot\|_{X_{0}+X_{1}})$ is also a Banach space (see, for instance, [4], lemma 2.3.1, p. 24).

An interpolation method (or functor) is a function $F$ that gives, for any pair $(X_{0},X_{1})$ in the class $\mathcal{C}$, a Banach space $F(X_{0},X_{1})$ so that

[TABLE]

and all the inclusions are continuous.

Given two compatible couples of Banach spaces $(X_{0},X_{1})$ and $(Y_{0},Y_{1})$, we will denote $f:(X_{0},X_{1})\rightarrow(Y_{0},Y_{1})$ to refer to a function $f:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}$ so that, for $j=0,1$, $f(X_{j})\subseteq Y_{j}$. All the theorems to appear will be interpreted differently depending on the interpolation method that will be taken into consideration.

Instead of studying properties of the spaces that arise when using an interpolation method, interpolation theory often tries to study the extent to which properties of linear functionals on the extremal spaces ($X_{0}$ and $X_{1}$) are maintained in the interpolated spaces (that is, the spaces that appear when the interpolation method is applied). In the second half of the 20th century, some authors started to consider other properties of functions, obtaining interpolation results for invertible functions or for compact linear operators, which is the focus of the results to come. The question that will interest us is to what extent compactness for a linear operator on one of the extremal spaces ($X_{0}$ or $X_{1}$) is enough to guarantee compactness of the operator on the interpolated spaces.

Despite the amount of work this question has motivated, it remains unsolved. Yet, it keeps attracting the attention of mathematicians, and some partial answers have been given. One of the first results on interpolation of compact operators dealt with the concrete case of $L_{p}$ spaces (Krasnolsel’skii, [23]), and in 1964 the first abstract results of this kind appeared (Lions and Peetre, [24], and Calderón, [5]). Cwikel studied the problem in the particular case where the interpolation method that is considered is the classical real interpolation method (see, for example, [12]. The interested reader can also refer to the papers by F. Cobos, D.E. Edmunds and A.J.B. Potter, [8], or Cobos and Peetre, [10]).

With respect to the complex interpolation method (which we will introduce in Definition 1.5), Cwikel, N. Krugljak and M. Mastylo proved in 1996 ([15]) that the problem of whether compactness of an operator between Banach couples extends to the interpolated spaces (if the operator is compact in one of the extremal spaces) can be reduced to the case where the spaces $Y_{0},\,Y_{1}$ and $X_{0}$ are reflexive and $X_{0}$ is compactly embedded into $Y_{0}$. In [14], Cwikel and Kalton completely solved the problem for the particular cases where the Banach couple $(X_{0},X_{1})$ is a couple of Banach lattices of measurable functions or when $X_{0}$ is a UMD-space (without extra conditions on the couple $(Y_{0},Y_{1})$). In 2010, Cwikel proved ([13]) that the compactness of the operator over the interpolated spaces is guaranteed when $(Y_{0},Y_{1})$ is a couple of complexified Banach lattices of measurable functions on a common measure space, if one of the following conditions is satisfied (without extra conditions on the couple $(X_{0},X_{1})$):

1. (1)

$Y_{0}$ or $Y_{1}$ has absolutely continuous norm, or 2. (2)

$Y_{0}$ and $Y_{1}$ have the Fatou property.

The results of this paper will focus not only on linear operators, but on homogeneous polynomials and on holomorphic functions (whose definition we will recall in Definition 1.2).

Before dealing with the definitions that we will need in the concrete topic of interpolation, we will need to introduce the theory of Fourier Series, in the more general setting of functions defined on Banach spaces:

Definition 1.1** ([14]).**

Let $X$ be a Banach space (over $\mathbb{C}$) and $f\in L^{2}(\mathbb{T},X)$, where $\mathbb{T}=\{|z|=1\}$. For $k$ in $\mathbb{Z}$, define the $k^{th}$** Fourier coefficient*** as*

[TABLE]

The Fourier Transform is defined to be

[TABLE]

and it is a linear isometry.

We will also make use of the theory of holomorphic functions defined on Banach spaces (over $\mathbb{C}$).

Definition 1.2** ([2]).**

Let $X$ and $Y$ be Banach spaces over $\mathbb{C}$ and let $f:X\rightarrow Y$ be a function. We say that $f$ is holomorphic if, for every $x\in X$, there exists a radius $r>0$ and a sequence of continuous polynomials $\{P_{m}f(x):X\rightarrow Y\}_{m=0}^{\infty}$ so that $P_{m}f(x)$ is $m-$homogeneous and, for every $y$ with $\|x-y\|_{X}<r$, we can write

[TABLE]

*where the convergence is uniform on every compact subset of $B^{\|\cdot\|_{X}}(x,r)$, or, equivalently, in one particular ball $B^{\|\cdot\|_{X}}(x,r^{\prime})$.

The series $\sum_{m=0}^{\infty}P_{m}f(x)(y-x)$ is called the Taylor series around $x$. We will also denote $\mathcal{H}(X;Y)=\{f:X\rightarrow Y\,:\,f\text{ is holomorphic}\}.$

Notice that the polynomials given by the Taylor series can be calculated by means of the $m^{th}$ Fréchet derivative

[TABLE]

For a holomorphic function $f:X\rightarrow Y$, we will denote the radius of convergence of $f$ at $x\in X$ as

[TABLE]

We will make use of [27], Theorem 12, where it is shown that $R^{c}(x)=R^{b}(x)=\sup\{r>0\,:\,f(B(x,r))\text{ is bounded}\}.$

Definition 1.3** ([2]).**

Let $X$ and $Y$ be two Banach spaces and $f:X\rightarrow Y$ be a holomorphic mapping. We say that $f$ is compact if, for every $x\in X$, there exists $r_{x}>0$ so that $f(B(x,r_{x}))$ is relatively compact in $Y$.

In [2], Proposition 3.4, the authors give a characterization of a compact holomorphic mapping in terms of the polynomials of its Taylor series. More concretely, it is shown that if $f:X\rightarrow Y$ is holomorphic, then it is compact if and only if ${d^{m}f(x)\over m!}$ is compact, for every $x$ in $X$ and natural number $m$.

Since we are dealing with polynomials, it will also be useful to recall the definition of the polar of a homogeneous polynomial:

Definition 1.4**.**

Let $P:X\rightarrow Y$ be a homogeneous polynomial of degree $m$. Then there exists a unique multilinear symmetric form, the polar of $P$, denoted by $\tilde{P}:\stackrel{{\scriptstyle m}}{{\overbrace{X\times\ldots\times X}}}\rightarrow Y$, so that $P(x)=\tilde{P}(\stackrel{{\scriptstyle m}}{{\overbrace{x,\ldots,x}}})$. Furthermore, the polarization identity gives a very precise formula for recovering the polar from the polynomial:

[TABLE]

*For a polynomial $P:X\rightarrow Y$, let us denote $\|P\|_{X\rightarrow Y}=\sup\{\|Px\|_{Y}\,:\,\|x\|_{X}\leq 1\}$ and, for a multilinear $n$-form $\|L\|_{X\rightarrow Y}=\sup\{\|L(x_{1},\ldots x_{n})\|_{Y}\,:\,\|x_{i}\|_{X}\leq 1,\,1\leq i\leq n\}$.

In the future, we will denote, for a Banach space $X$, $B_{X}=\{x\in X\,:\,\|x\|_{X}\leq 1\}$, if there is no mistake about what norm we are using.

Trivially, $\|P\|_{X\rightarrow Y}\leq\|\tilde{P}\|_{X\rightarrow Y}$. Reciprocally, Martin’s theorem ([16], Proposition 1.8, p. 10) allows us to write, for a $m$-homogeneous polynomial $P$,*

[TABLE]

We will denote by $\mathcal{P}(^{n}X)$ the space of all homogeneous polynomials of degree $n$ from $X$ to $\mathbb{C}$ and $\mathcal{L}(^{n}X)$ the space of all multilinear $n$-symmetric forms from $X$ to $\mathbb{C}$. Given two compatible couples of Banach spaces $(X_{0},X_{1}),\,(Y_{0},Y_{1})\in\mathcal{C}$ we will also denote by $\mathcal{P}(^{n}(X_{0},X_{1}),(Y_{0},Y_{1}))$ the space of all homogeneous polynomials $P:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}$ so that $P$ is homogeneous of degree $n$ and $P:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1},\,P:X_{j}\rightarrow Y_{j}\,(j=0,1)$ are continuous.

Calderón asked in [5] the following analogous problem for bilinear operators: if we are given compatible couples of Banach spaces $(X_{0}^{(1)},X_{1}^{(1)}),\,(X_{0}^{(2)},X_{1}^{(2)})$ and $(Y_{0},Y_{1})$ and we consider a bilinear operator

[TABLE]

that satisfies that $T(X_{0}^{(j)}\times X_{1}^{(j)})\subseteq Y_{j}$ is bounded for $j=0,1$ and $T_{|X_{0}\times X_{0}}$ is compact, can we guarantee that $T_{F(X_{0}\times X_{0},X_{1}\times X_{1})}$ is compact, for a certain interpolation functor $F$? We refer to [4], page 96, for the corresponding definitions for multilinear operators. Fernández and da Silva ([18]) studied some particular cases under the real method, and recently in 2017, Fernández-Cabrera and Martínez ([20]) studied how the real method worked with a function parameter, and studied also the complex method. We would like to stress that, even though we will be working with polynomials, the questions concerning compact multilinear operators and compact polynomials will be analogous.

Let next $\mathbb{L}$ denote the set $\{z\in\mathbb{C}\,:\,1<|z|<e\}$ and, for a compatible couple of Banach spaces $(X_{0},X_{1})$, define the function space $\mathcal{F}\{X_{0},X_{1}\}$ as follows:

[TABLE]

This space is a Banach space if given the norm

[TABLE]

Notice that, if $\varphi\in\mathcal{F}\{X_{0},X_{1}\}$, then automatically $\varphi\in L^{2}(\mathbb{T},X_{0}+X_{1})$.

Definition 1.5** ([14]).**

Define the Calderón Complex Interpolation method as the functor that associates to each value $0<\theta<1$ the intermediate space

[TABLE]

which is a Banach space if endowed with the norm

[TABLE]

In the following, if there is no confusion about which spaces are to be interpolated, we will denote $\mathbf{X}_{\theta}=[X_{0},X_{1}]_{\theta}$.

Definition 1.6** ([14]).**

For each $0<\theta<1$, define the Peetre Interpolation method as the method which proceeds as follows: For each value $0<\theta<1$,

[TABLE]

This space is a Banach space, endowed with the norm

[TABLE]

where the supremum is taken over all complex valued $(\lambda_{k})_{k=-\infty}^{\infty}$ with $|\lambda_{k}|\leq 1$ for all $k$, and the infimum is taken over all representations $x=\sum_{k\in\mathbb{Z}}x_{k}$ as above.

With the Calderón complex method, we have the following classical Interpolation Theorem, due to Riesz and Thorin:

Theorem 1.7** (M. Riesz, G.O. Thorin, [4], p. 2).**

Let $0<p_{i},q_{i}<\infty,\,0<\theta<1$. Assume $p_{0}\neq p_{1}$ and $q_{0}\neq q_{1}$ and define $p,\,q$ by

[TABLE]

Assume that

[TABLE]

*are linear operators bounded by $M_{0}$ and $M_{1}$, respectively.

Then,*

[TABLE]

*is bounded and continuous with norm $M\leq M_{0}^{1-\theta}M_{1}^{\theta}$.

We remark that if $d\mu=d\nu$ we have $[L_{p_{0}}(A_{0}),L_{\infty}(A_{1})]_{\theta}=L_{p}([A_{0},A_{1}]_{\theta})$, for every compatible couple of Banach spaces $A_{0},\,A_{1}$.*

More generally, we will make use of the following generalization of the Interpolation Theorem for linear operators, which can be consulted in [4] (theorem 4.4.1, p. 96):

Theorem 1.8**.**

Let $(X_{0}^{(j)},X_{1}^{(j)})_{j=1}^{n},\,(Y_{0},Y_{1})$ be compatible couples of Banach spaces. Assume that

[TABLE]

*is an $n-$multilinear form and satisfies $\|T(x_{1},\ldots,x_{n})\|_{Y_{j}}\leq M_{j}$, for $x_{l}\in B_{X_{j}^{(l)}}$, $1\leq l\leq n,\,j=0,1$.

Then, for any $0<\theta<1$, $T$ can be uniquely extended to a multilinear mapping*

[TABLE]

with $\|T\|_{E\rightarrow[Y_{0},Y_{1}]_{\theta}}\leq M_{0}^{1-\theta}M_{1}^{\theta}$.

Calderón’s method behaves very well with procedures like reiteration of the interpolation method. More concretely, the result below can be also found in [4], Theorem 4.2.2, p. 91:

Theorem 1.9**.**

Let $0\leq\theta\leq 1$. Then

1. a)

$X_{0}\cap X_{1}$* is dense in $[X_{0},X_{1}]_{\theta}$ (using the $\|\cdot\|_{\theta}$-norm).* 2. b)

[TABLE] 3. c)

The space $[X_{0},X_{1}]_{j}$ is a closed subspace of $X_{j}$ for $j=0,1$, with equality of norms. 4. d)

[TABLE]

We will make use of the following result, proved by Lions and Peetre ([24], ch. IV, Theorem 1.1, p. 29):

Theorem 1.10**.**

Let $X_{0},\,X_{1}$ be Banach spaces and $0<\theta<1$. Then,

1. (1)

there exists a constant $C>0$ so that, for every $x\in X_{0}\cap X_{1}$,

[TABLE] 2. (2)

There exists a constant $C^{\prime}>0$ so that, for every $x\in[X_{0},X_{1}]_{\theta}$ and $t>0$, one can find $x_{0}\in X_{0}$ and $x_{1}\in X_{1}$ satisfying:

[TABLE]

The results of this paper are presented as follows: In section 2 we will give an answer to the natural question of whether a function which is holomorphic as a function between $X_{0}+X_{1}$ and $Y_{0}+Y_{1}$, and as a function between $X_{j}$ and $Y_{j}$ ($j=0,1$) is holomorphic as well when restricted to $[X_{0},X_{1}]_{\theta}$, $0\leq\theta\leq 1$. That is, we will proof an analogous theorem to the Riesz-Thorin Theorem, but for the more general setting of holomorphic functions instead of linear operators. This theorem will be of special importance for us if we want to reduce the study of compactness between the interpolated spaces to the study of the compactness of the polynomials that appear in the Taylor series, in virtue of Proposition 3.4 from [2].

In the same section, we will follow the ideas suggested in [14] by Cwikel and Kalton to to prove some preliminary technical lemmas. In section 3, we will continue with the ideas from [14] to prove a theorem about compactness on the interpolated spaces, if in the domain space we use Peetre’s interpolation space and in the range space we consider Calderón’s interpolation method. Some of the procedures Cwikel and Kalton carried out for linear operators have an analogous application for polynomials, since linearity was not especially employed in the proofs. Some other results display a very strong dependance on the linearity of the considered operator, and we will be required to reach similar conclusions through other techniques.

In section 4 we will focus on some classic results. We will also prove a classical polarization-like proposition (Lemma 4.5), which we believe is of interest beyond Interpolation Theory.

2. A theorem about interpolation of holomorphic functions and some supporting lemmas

Before stating the corresponding theorem for holomorphic functions, let us prove an interpolation result for continuous functions.

Proposition 2.1**.**

*Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be two couples of Banach spaces and let $f:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}$ be a holomorphic function. Assume that $f:X_{i}\rightarrow Y_{i}$ is continuous, for $i=0,1$. Let $0\leq\theta\leq 1$.

Then, $f:[X_{0},X_{1}]_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is continuous.*

Proof.

Let $x\in[X_{0},X_{1}]_{\theta}$. Then, we can find $\varphi\in\mathcal{F}\{X_{0},X_{1}\}$ with $x=\varphi(e^{\theta})$.

Let $\varepsilon>0$ and $t\in[0,2\pi]$. Since $f:X_{0}\rightarrow Y_{0}$ is continuous, we can find $\delta_{t}^{0}>0$ so that, if $\|\zeta-\varphi(e^{it})\|_{X_{0}}<\delta_{t}^{0}$, then $\|f(\zeta)-f\circ\varphi(e^{it})\|_{Y_{0}}<{\varepsilon\over 2}$. Using continuity of $\varphi:\{z\in\mathbb{C}\,:\,|z|=1\}\rightarrow X_{0}$, we can find $t_{1},\ldots,t_{N}\in[0,2\pi]$ with

[TABLE]

Notice that, if \varphi(e^{it})\in B\Big{(}\varphi(e^{it_{k}});{\delta_{t_{k}}^{0}\over 2}\Big{)} and $\|\zeta-\varphi(e^{it})\|_{X_{0}}<{\delta_{t_{k}}^{0}\over 2}$, then $\|\zeta-\varphi(e^{it_{k}})\|_{X_{0}}<\delta_{t_{k}}^{0}$ and

[TABLE]

Analogously, we can find $t^{(1)},\ldots,t^{(L)}\in[0,2\pi]$ and $\delta^{1}_{t^{(1)}},\ldots,\delta^{1}_{t^{(L)}}>0$ so that

[TABLE]

and, if \varphi(e^{1+it})\in B\Big{(}\varphi(e^{1+it^{(j)}};{\delta^{1}_{t^{(j)}}\over 2}\Big{)} and $\|\zeta-\varphi(e^{1+it})\|_{X_{1}}<{\delta^{1}_{t^{(j)}}\over 2}$, then

[TABLE]

Choose $\delta=\min\left\{{\delta_{t_{k}}^{0}\over 2},{\delta_{t^{(j)}}^{1}\over 2}\right\}_{k,j=1}^{N,L}>0$ and consider $\zeta\in[X_{0},X_{1}]_{\theta}$ so that $\|\zeta-x\|_{[X_{0},X_{1}]_{\theta}}<\delta$.

Then, we can find $\psi\in\mathcal{F}\{X_{0},X_{1}\}$ with $\psi(e^{\theta})=\zeta-x$ and

[TABLE]

Also, notice that $f(\zeta)-f(x)=f\circ(\psi+\varphi)(e^{\theta})-f\circ\varphi(e^{\theta})=[f\circ(\psi+\varphi)-f\circ\varphi](e^{\theta})$.

Let $t\in[0,2\pi]$. Then, we can find $1\leq t_{k_{0}}\leq N$ with \varphi(e^{it})\in B\Big{(}\varphi(e^{it_{k_{0}}});{\delta_{t_{k}}^{0}\over 2}\Big{)}. Now,

[TABLE]

so $\|f\circ(\psi+\varphi)(e^{it})-f\circ\varphi(e^{it})\|_{Y_{0}}<\varepsilon$.

Analogously, $\|f\circ(\psi+\varphi)(e^{1+it})-f\circ\varphi(e^{1+it})\|_{Y_{1}}<\varepsilon$. In conclusion,

[TABLE]

for every $\|\zeta-x\|_{[X_{0},X_{1}]_{\theta}}<\delta$, and the result follows.

∎

Remark 2.2**.**

The hypothesis of $f:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}$ being holomorphic in Theorem 2.1 is nothing more than a technicality to guarantee that $f:[X_{0},X_{1}]_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is well-defined since, by definition, an element $y\in[Y_{0},Y_{1}]_{\theta}$ must be of the form $y=\psi(e^{\theta})$, with $\psi\in\mathcal{F}\{Y_{0},Y_{1}\}$. To guarantee that $f\circ\varphi\in\mathcal{F}\{Y_{0},Y_{1}\}$ for $\varphi\in\mathcal{F}\{X_{0},X_{1}\}$, it is indeed enough to assume that $f:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}$ is holomorphic and $f:X_{i}\rightarrow Y_{i}$ is continuous, for $i=0,1$. Precisely because of this, this theorem is the most general for continuity over the interpolated spaces that can be enunciated for Calderón’s complex method.

Having Theorem 2.1 at hand, we can now focus on the question of whether the property of holomorphy can be obtained when restricted to the interpolated spaces via the Calderón’s method, taking into account the considerations collected in Remark 2.2.

Theorem 2.3**.**

*Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be two couples of Banach spaces and let $f:\mathbf{X}\rightarrow\mathbf{Y}$ be a function so that $f:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}$ and $f:X_{j}\rightarrow Y_{j}$ are holomorphic $(j=0,\,1)$.

Then, $f:\mathbf{X}_{\theta}\rightarrow\mathbf{Y}_{\theta}$ is holomorphic, for every $0\leq\theta\leq 1$.*

Proof.

Let first $x\in X_{0}\cap X_{1}$. Then, we know there exists a sequence of homogeneous polynomials, given by \big{\{}{d^{m}f(x)\over m!}\big{\}}_{m=0}^{\infty}, so that

[TABLE]

where the convergence occurs uniformly on $B^{\|\cdot\|_{j}}(x;r)$ for $0<r<R_{j}(x)$, $j=0,1$.

Furthermore, we know

[TABLE]

Let us define $P_{m}f(x)={d^{m}f(x)\over m!}$ and let $\tilde{P}_{m}f(x)$ be as in Theorem 1.4. Use the Polarization Constant and Martin’s theorem ([25]), together with Theorem 1.8, to write

[TABLE]

Hence, taking also into account Stirling’s formula,

[TABLE]

Therefore,

[TABLE]

and hence

[TABLE]

uniformly for $\zeta\in B^{\|\cdot\|_{\mathbf{X}_{\theta}\rightarrow\mathbf{Y}_{\theta}}}(x;r)$, $0<r<R_{\theta}(x)$.

Let now $x\in{\mathbf{X}}_{\theta}$. From Theorem 1.9 (a)) we can find a sequence $\{x_{n}\}_{n=1}^{\infty}\subseteq X_{0}\cap X_{1}$ so that

[TABLE]

Since $f:{\mathbf{X}}_{\theta}\rightarrow{\mathbf{Y}}_{\theta}$ is continuous at $x$ (because of Proposition 2.1), we can find $\delta>0$ so that, if $\|x-\zeta\|_{\mathbf{X}_{\theta}}<\delta$, then $\|f(x)-f(\zeta)\|_{\mathbf{Y}_{\theta}}<1$. Let $n_{0}\in\mathbb{N}$ so that $\|x-x_{n_{0}}\|_{\mathbf{X}_{\theta}}<{\delta\over 3}$. Then, $f$ is bounded for $\|\zeta-x_{n_{0}}\|_{\mathbf{X}_{\theta}}<{\delta\over 2}$. Since for a holomorphic function the radius of convergence of the Taylor series coincides with the radius of boundedness of the function, we can write

[TABLE]

uniformly for $\|\zeta-x_{n_{0}}\|_{\mathbf{X}_{\theta}}<{\delta\over 2}$. Then, the convergence of the series will also happen uniformly for $\|x-\zeta\|_{\mathbf{X}_{\theta}}\leq r$, for every $r<{\delta\over 6}$.

∎

We recall the holomorphic analogue of adjoint operator:

Definition 2.4** ([2]).**

Let $f:X\rightarrow Y$ be a holomorphic mapping. We define its adjoint as

[TABLE]

Notice that in the case of $f$ being a polynomial (resp. a multilinear mapping) we obtain $f^{*}:Y^{*}\rightarrow\mathcal{P}(^{n}X)$ (resp. $f^{*}:Y^{*}\rightarrow\mathcal{L}(^{n}X)$), and in those cases we can write $\|f\|=\|f^{*}\|$. Also, notice ([2]) that $f$ is a compact mapping if and only if $f^{*}$ is also compact (for the case of multilinear mappings, just apply the polarization formula).

Taking this into account, we can prove the following lemma, analogous to a well-known classical result.

Lemma 2.5** (Riemann-Lebesgue Lemma).**

Let $X$ and $Y$ be Banach spaces, $f\in L^{2}(\mathbb{T},X)$ and let $P:X\rightarrow Y$ be a compact polynomial. Then, $\|\widehat{Pf}(k)\|_{Y}\xrightarrow[|k|\rightarrow\infty]{\,}0$.

Proof.

Given $y^{*}\in B_{Y^{*}}$ and using that $\mathfrak{F}:(L^{2}(\mathbb{T},X))\rightarrow\ell^{2}(X)$ is an isometry, we know

[TABLE]

so that $|y^{*}\widehat{Pf}(k)|\xrightarrow[|k|\rightarrow\infty]{\,}0$.

Let $\varepsilon>0$ and find $y_{1}^{*},\ldots,y_{N}^{*}\in B_{Y^{*}}$ so that $P^{*}B_{Y^{*}}\subseteq\cup_{j=1}^{N}B(P^{*}y_{j}^{*};\varepsilon)$. Also, let $k_{\varepsilon}\in\mathbb{N}$ so that for every $|k|\geq k_{\varepsilon}$, $\max_{1\leq j\leq N}|y_{j}^{*}\widehat{Pf}(k)|<\varepsilon$. Then, if $y^{*}\in B_{Y^{*}}$, there exists $1\leq j_{0}\leq N$ such that

[TABLE]

Therefore, if $|k|\geq k_{\varepsilon}$, $|y^{*}\widehat{Pf}(k)|\leq|y^{*}\widehat{Pf}(k)-y_{j_{0}}^{*}\widehat{Pf}(k)|+|y_{j_{0}}^{*}\widehat{Pf}(k)|<\varepsilon(1+\|f\|_{L^{2}})$.

In conclusion, for every $y^{*}\in B_{Y^{*}}$ and $|k|\geq k_{\varepsilon}$,

[TABLE]

∎

We remark that we have used in a very concrete way the fact that $P$ is compact, and that the proof, as outlined above, does not work for an arbitrary polynomial.

Lemma 2.6**.**

Let $P:X\rightarrow Y$ be a compact homogeneous polynomial of degree $l$ and $\tilde{P}:\stackrel{{\scriptstyle l}}{{\overbrace{X\times\ldots\times X}}}\rightarrow Y$ be its polar. Suppose $\{f_{n}^{(j)}\}_{n=1}^{\infty}$ are bounded sequences in $L^{2}(\mathbb{T},X)$, $j=1,\ldots,l$ and that, for some $k$ in $\mathbb{N}$,

[TABLE]

*for every $y^{*}$ in $B_{Y^{*}}$.

Then,*

[TABLE]

Proof.

Without loss of generality, assume that the sequences $\{f_{n}^{(j)}\}_{n=1}^{\infty}$ are bounded by $1$ and let $\{y_{m}^{*}\}_{m=1}^{\infty}\subseteq B_{Y^{*}}$ be such that $\{(\tilde{P}^{*})(y_{m}^{*})\}_{m=1}^{\infty}$ is dense in $(\tilde{P}^{*})(B_{Y^{*}})$. Then, any $l-$tuples of bounded sequences $\{\zeta_{n}^{(1)}\},\ldots,\{\zeta_{n}^{(l)}\}$ for which $\lim_{n\rightarrow\infty}y_{m}^{*}\tilde{P}(\zeta_{n}^{(1)},\ldots,\zeta_{n}^{(l)})=0$ for every $m$ must satisfy $\lim_{n\rightarrow\infty}y^{*}\tilde{P}(\zeta_{n}^{(1)},\ldots,\zeta_{n}^{(l)})=0$ for all $y^{*}$ in $B_{Y^{*}}$. Applying compactness,

[TABLE]

Let us show that then we can then conclude that given $\varepsilon>0$ there is a constant $C>0$ so that

[TABLE]

for every $x^{(1)},\ldots,x^{(l)}\in X$.

Indeed, otherwise there exists $\varepsilon>0$ such that, for every $n$ in $\mathbb{N}$, we can find $x_{n}^{(1)},\ldots,x_{n}^{(l)}$ in $X$ with

[TABLE]

Defining $\zeta_{n}^{(j)}={x_{n}^{(j)}\over\|x_{n}^{(j)}\|_{X}}\in B_{X}$ we obtain, for $m$ in $\mathbb{N}$,

[TABLE]

so that $\lim_{n\rightarrow\infty}|y_{m}^{*}(\tilde{P}(\zeta_{n}^{(1)},\ldots,\zeta_{n}^{(l)}))|=0$ and, therefore, $\lim_{n\rightarrow\infty}\|\tilde{P}(x_{n}^{(1)},\ldots,x_{n}^{(l)})\|^{k}_{Y}=0$, contradicting $\|\tilde{P}(\zeta_{n}^{(1)},\ldots,\zeta_{n}^{(l)}))\|_{Y}^{k}>\varepsilon$ for every $n$.

Finally, given $\varepsilon>0$, just write

[TABLE]

and the result follows. ∎

In the results to come, we shall use the following lemma, whose proof can be found in [14], Lemma 2-(i):

Lemma 2.7**.**

Let $X_{0},\,X_{1}$ be a compatible couple of Banach spaces. For each $0<\theta<1$ there is a constant $C=C(\theta)$ such that, for every $\varphi\in\mathcal{F}\{X_{0},X_{1}\}$,

[TABLE]

In particular, for all $x\in X_{0}\cap X_{1}$,

[TABLE]

Definition 2.8**.**

Let $X$ be a Banach space. For $N\in\mathbb{N}$ and $f\in\mathcal{H}(X,\mathbb{C})$, define the functions $S_{N}f$ as follows:

[TABLE]

By the uniform $L_{1}$-boundedness of the de la Vallée Poussin kernels, if $(X_{0},X_{1})$ is a compatible couple of Banach spaces, there exists a constant $C$ such that $\|S_{N}\varphi\|_{\mathcal{F}\{X_{0},X_{1}\}}\leq C\|\varphi\|_{\mathcal{F}\{X_{0},X_{1}\}}$ for every $\varphi\in\mathcal{F}\{X_{0},X_{1}\}$, $N>0$ (for further details, check the comments in [14] and the references therein).

The next lemma will provide some crucial tools for the proof of the main theorem in the next section:

Lemma 2.9**.**

Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be two compatible couples of Banach spaces. Let also $l\in\mathbb{N}$ and $P\in\mathcal{P}(^{l}\mathbf{X};\mathbf{Y})$ so that $P:X_{0}\rightarrow Y_{0}$ is compact. Then,

1. (1)

The set $\left\{\widehat{P\varphi}(k)\,:\,\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}},\,k\in\mathbb{Z}\right\}$ is relatively compact in $Y_{0}$. 2. (2)

$\displaystyle\lim_{|k|\rightarrow\infty}\sup\left\{\|\widehat{P\varphi}(k)\|_{Y_{0}}\,:\,\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}}\right\}=0$. 3. (3)

For each $\delta>0$, there exists $L=L(\delta)$ so that, for every $\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}}$,

[TABLE] 4. (4)

For each $0<\theta<1$, we have

[TABLE]

Proof.

hola

1. (1)

First of all, notice that, again since $\mathfrak{F}$ (the Fourier transform) is a linear isometry from $L^{2}(\mathbb{T},X_{0}+X_{1})$ to $\ell^{2}(X_{0}+X_{1})$ and $P$ is a compact polynomial, it follows that $\mathfrak{F}P$ is a compact operator and hence $\left\{(\widehat{P\varphi}(k))_{k=-\infty}^{\infty}\,:\,\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}}\right\}\subseteq Y_{0}^{\mathbb{Z}}$ is a relatively compact subset (considering the $\|\cdot\|_{2}$-norm and, as a consequence, in the sup norm).

Now, applying lemma 2.5, we find that in fact we have

[TABLE]

Claim: If $W$ is a Banach space and K\subseteq\big{(}c_{0}(W),\|\cdot\|_{\infty}\big{)} is compact, then $\{k(n)\,:\,k\in K,\,n\in\mathbb{Z}\}$ is a relatively compact subset of $W$.

Indeed, assume $\{k_{j}\}_{j=1}^{\infty}\subseteq K$ and $\{m_{j}\}_{j=1}^{\infty}\subseteq\mathbb{Z}$. Then, we know there exists $\{n_{j}\}_{j=1}^{\infty}\subseteq\mathbb{N}$ so that $\|k_{n_{j}}-k_{n_{l}}\|_{\infty}\xrightarrow[j,l\rightarrow\infty]{\,}0$. Let us show that $\{k_{n_{j}}(m_{n_{j}})\}$ converges.

Let $\varepsilon>0$. Then, there exists $j_{0}$ such that, for every $j,l\geq j_{0}$ $\|k_{n_{j}}-k_{n_{l}}\|_{\infty}<{\varepsilon\over 3}$. Let next $j_{\varepsilon}\geq j_{0}$ so that $\|k_{n_{j_{0}}}(m_{n_{j}})\|_{W}<{\varepsilon\over 6}$ for every $j\geq j_{\varepsilon}$, and take $j,l\geq j_{\varepsilon}$. Then

[TABLE] 2. (2)

Let us prove a more general result, namely that if $\{x_{j}\}_{j\in J}\subseteq c_{0}(Y_{0})$ is a compact subset, then

[TABLE]

Indeed, assume otherwise that we can find $\varepsilon>0$ such that, for every natural number $n$ there exists $N_{n}\geq n$ and $j_{N_{n}}\in J$ so that $\|x_{j_{N_{n}}}(N_{n})\|_{Y_{0}}\geq\varepsilon$. Without loss of generality, we can assume that $\{N_{n}\}_{n=1}^{\infty}$ is an increasing sequence.

Now, by compactness, we can find $j_{0}\in J$ and a subsequence (which, to simplify the notation, we will still denote by $\{x_{j_{N_{n}}}\}$) so that

[TABLE]

Hence, given ${\varepsilon\over 2}>0$, there exists a natural number $n_{\varepsilon}$ so that, for every $n\geq n_{\varepsilon}$ we can guarantee $\|x_{j_{0}}-x_{j_{N_{n}}}\|_{\infty}<{\varepsilon\over 2}$. In other words, for every $n\geq n_{\varepsilon}$ and every integer $k$ we have

[TABLE]

In particular, for every $n\geq n_{\varepsilon}$ we have

[TABLE]

which contradicts $x_{j_{0}}\in c_{0}(Y_{0})$. The argument for the case when $n\rightarrow-\infty$ follows in the same way. 3. (3)

Using compactness of $P^{*}$ we can find $y_{1}^{*},\ldots,y_{N}^{*}$ in $B_{Y_{0}^{*}}$ so that

[TABLE]

Therefore, if $y^{*}\in B_{Y^{*}_{0}}$, there exists $1\leq j_{0}\leq N$ so that, for every $x\in B_{X_{0}}$,

[TABLE]

Hence, for every $k\in\mathbb{Z}$,

[TABLE]

From this

[TABLE]

Let next $\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}}$ and define $A^{(\varphi)}_{\delta}=\{k\in\mathbb{Z}\,:\,\|\widehat{P\varphi}(k)\|_{Y_{0}}\geq\delta\}$. Then, for each $k\in A^{(\varphi)}_{\delta}$,

[TABLE]

Apply Parseval’s Identity (2.1) to obtain

[TABLE]

for every $\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}}$. 4. (4)

First, compute

[TABLE]

using Lemma 2.7 and the change $u=i+t$, for every $k\in\mathbb{Z}$.

Therefore, using part 2, we can conclude $\|\widehat{P\varphi}(k)e^{k\theta}\|_{\mathbf{Y}_{\theta}}\xrightarrow[|k|\rightarrow\infty]{\,}0$.

Assume next (in order to simplify the notation) that $\|P\|_{X_{0}\rightarrow Y_{0}}\leq 1$, so that $\|\widehat{P\varphi}(k_{n})\|_{Y_{0}}\leq 1$ for every $\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}}$ and $n\in\mathbb{N}$. Assume also that there exists $\delta>0$, $\{k_{n}\}\searrow-\infty$ and $\{\varphi_{n}\}\subseteq B_{\mathcal{F}\{X_{0},X_{1}\}}$ so that $k_{n}<2k_{n-1}$ and $\|\widehat{P\varphi_{n}}(k_{n})e^{k_{n}\theta}\|_{\mathbf{Y}_{\theta}}\geq\delta$.

Now, given $n$ and $\varepsilon>0$, we can use part 1 to find $m,\,p\in\mathbb{N}$ so that $m>p\geq n$ and $\|\widehat{P\varphi}_{m}(k_{m})-\widehat{P\varphi}_{p}(k_{p})\|_{Y_{0}}<\varepsilon$.

Also, $\|\widehat{P\varphi}_{m}(k_{m})e^{k_{m}}\|_{Y_{1}}\leq 1$ and $\|\widehat{P\varphi}_{p}(k_{p})e^{k_{m}}\|_{Y_{1}}\leq e^{k_{m}-k_{p}}\leq 1$. Then,

[TABLE]

Therefore,

[TABLE]

which is a contradiction to the assumption $\|\widehat{P\varphi}_{n}(k_{n})e^{k_{n}\theta}\|_{\mathbf{Y}_{\theta}}\geq\delta>0$.

∎

Lemma 2.10**.**

Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be two compatible couples of Banach spaces and let $l\in\mathbb{N}$ and $P\in\mathcal{P}(^{l}\mathbf{X};\mathbf{Y})$ so that $P:X_{0}\rightarrow Y_{0}$ is compact. For $0<\theta<1$ and $N\in\mathbb{N}$, the set \{S_{N}\big{(}P\varphi\big{)}(e^{\theta})\,:\,\varphi\in B_{\mathcal{F}\{X_{0},X_{1}\}}\} is relatively compact in $\mathbf{Y}_{\theta}$.

Proof.

Suppose $\{\varphi_{n}\}_{n=1}^{\infty}\subseteq B_{\mathcal{F}\{X_{0},X_{1}\}}$. By lemma 2.9, part 1, we can pass to a subsequence $\{\psi_{n}\}$ such that, for every $|k|\leq 2N$,

[TABLE]

for all $n\in\mathbb{N}$. Then,

[TABLE]

for $|z|=1$ and all natural numbers $n$. Also, for $|z|=e$, we have \|S_{N}\big{(}P\psi_{n}\big{)}(z)-S_{N}\big{(}P\psi_{n+1}\big{)}(z)\|_{Y_{1}}\leq C_{1}, for some suitable constant $C_{1}>0$ and natural number $n$.

Thus, by lemma 2.7,

[TABLE]

so \{S_{N}\big{(}P\psi_{n}\big{)}(e^{\theta})\}_{n=1}^{\infty} is convergent.

∎

Lemma 2.11**.**

Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be two compatible couples of Banach spaces and let $\mho$ be a subset of $B_{\mathcal{F}\{X_{0},X_{1}\}}$. Choose $0<\theta<1$ and define $\mho_{\theta}=\{\varphi(e^{\theta})\,:\,\varphi\in\mho\}$. Assume that every sequence $\{\varphi_{n}\}\subseteq\mho$ satisfies

[TABLE]

Then, $P(\mho_{\theta})$ is relatively compact in $\mathbf{Y}_{\theta}$.

Proof.

First of all, we remark that the hypotheses imply that \lim_{n\rightarrow\infty}\|P\varphi(e^{\theta})-S_{n}\big{(}P\varphi\big{)}(e^{\theta})\|_{\mathbf{Y}_{\theta}}=0 uniformly for $\varphi\in\mho$. Indeed, otherwise we can find $\varepsilon>0$ and two subsequences, $\{N_{n}\}_{n=1}^{\infty}\subseteq\mathbb{N}$ and $\{\varphi_{N_{n}}\}_{n=1}^{\infty}\subseteq\mho$, such that \|P\varphi_{N_{n}}(e^{\theta})-S_{N_{n}}\big{(}P\varphi_{N_{n}}\big{)}(e^{\theta})\|_{\mathbf{Y}_{\theta}}\geq\varepsilon, which contradicts the assumptions.

Let now $\{\varphi_{n}\}\subseteq\mho$. From Lemma 2.10 we know that \{S_{1}\big{(}P\varphi_{n}\big{)}(e^{\theta})\} contains a subsequence \{S_{1}\big{(}P\varphi_{n_{k,1}}\big{)}(e^{\theta})\}_{k=1}^{\infty} which is convergent in $\mathbf{Y}_{\theta}$. Inductively, assume we have obtained a subsequence $\{\varphi_{n_{k,j}}\}_{k=1}^{\infty}$ so that \{S_{l}\big{(}P\varphi_{n_{k,j}}\big{)}(e^{\theta})\}_{j=1}^{\infty} converges, for $1\leq l\leq j$. Then, using again Lemma 2.10, \{S_{j+1}\big{(}P\varphi_{n_{k,j}}\big{)}(e^{\theta})\}_{k=1}^{\infty} contains a subsequence \{S_{j+1}\big{(}P\varphi_{n_{k,j+1}}\big{)}(e^{\theta})\}_{k=1}^{\infty} which is convergent in $\mathbf{Y}_{\theta}$.

Choose then $\{\varphi_{n_{k,k}}\}_{k=1}^{\infty}$ and let us show that $\{P\varphi_{n_{k,k}}(e^{\theta})\}$ is a Cauchy sequence. Indeed, let $\varepsilon>0$. Then, we know there exists $N\in\mathbb{N}$ so that, for every $n\geq N$,

[TABLE]

for every $k\geq 1$.

Then, since \{S_{N}\big{(}P\varphi_{n_{k,N}}\big{)}(e^{\theta})\}_{k=1}^{\infty} is convergent, there exists $\tilde{N}\geq N$ so that, if $l,m\geq\tilde{N}$,

[TABLE]

Let $N_{\varepsilon}=\tilde{N}$ and $i,j\geq N_{\varepsilon}$. Then, $n_{i,i}=n_{l,N}$ and $n_{j,j}=n_{m,N}$ for some $l,m\geq\tilde{N}$. Hence,

[TABLE]

∎

3. An interpolation result for compact holomorphic functions, by the methods of Cwikel and Kalton.

The main result appears as a corollary to Theorem 3.1, which is itself supported by the lemmas presented in the second half of Section 2.

Theorem 3.1**.**

*Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be compatible couples of Banach spaces. Let $l\in\mathbb{N}$ and $P\in\mathcal{P}(^{l}\mathbf{X};\mathbf{Y})$ so that $P:X_{0}\rightarrow Y_{0}$ is compact and let $\mho$ be the subset of $B_{\mathcal{F}\{X_{0},X_{1}\}}$ consisting of those elements $\varphi$ for which the series $\sum_{k\in\mathbb{Z}}e^{jk}\hat{\varphi}(k)$ converges unconditionally in $X_{j}$ and $\left\|\sum_{k\in\mathbb{Z}}\lambda_{k}e^{jk}\hat{\varphi}(k)\right\|_{X_{j}}<1$ for $j=0,\,1$ and for every sequence of complex scalars $\{\lambda_{k}\}$ so that $|\lambda_{k}|<1$ for all $k$.

Then, $P(\mho_{\theta})$ is relatively compact in $\mathbf{Y}_{\theta}$ for every $0<\theta<1$.*

Proof.

Let $0<\theta<1$ and assume (without loss of generality and to simplify the calculations) that $\|P\|_{\mathbf{X}_{\theta}\rightarrow\mathbf{Y}_{\theta}}\leq 1$. Let $\{\varphi_{n}\}\subseteq\mho$. It will suffice to show $\lim_{n\rightarrow\infty}\|P\varphi_{n}(e^{\theta})\|_{\mathbf{Y}_{\theta}}=0$. Indeed, using Lemma 2.10 we can conclude that

[TABLE]

so we would be in the situation of Lemma 2.11 and the result would follow. To simplify the notation, assume without loss of generality that $\widehat{P\varphi}_{n}(k)=0$ for $n\in\mathbb{N}$ and $|k|\leq n$.

For any $N\in\mathbb{N}$ let us pick a subset $A_{n}(N)\subseteq\mathbb{Z}$ so that $\text{card}(A_{n}(N))=N$ and $\|\widehat{P\varphi}_{n}(k)\|_{\mathbf{Y}_{\theta}}\leq\|\widehat{P\varphi}_{n}(l)\|_{\mathbf{Y}_{\theta}}$ whenever $k\notin A_{n}(N)$ and $l\in A_{n}(N)$. We may use Lemma 2.9, part 4 to see that for any fixed $N$ we must have

[TABLE]

It is therefore possible to pick a non-decreasing sequence of integers $N_{n}$ with $\lim_{n\rightarrow\infty}N_{n}=\infty$ so that

[TABLE]

We need to deal now with $\sum_{k\notin A_{n}(N_{n})}\widehat{P\varphi}_{n}(k)z^{k}$. Let $b_{n}=\sup_{k\in\mathbb{Z}}\|\widehat{P\varphi}_{n}(k)\|_{\mathbf{Y}_{\theta}}$. Then, by Lemma 2.9, part 3, we have $\lim_{n\rightarrow\infty}b_{n}=0$.

Let $y^{*}\in B_{Y_{0}^{*}}\cap B_{Y_{1}^{*}}$. Then,

[TABLE]

Now, notice that, if $T$ is a bilinear form (for the sake of simplification of the notation, we will assume $\|T\|\leq 1$),

[TABLE]

taking into consideration the fact that $\varphi_{n}\in\mho$. We remark that the steps for proving equation (3.2) can be followed in order to obtain the analogous result for an $s$-multilinear form $T$,

[TABLE]

making the corresponding changes.

Therefore, we can deduce that

[TABLE]

On the other hand,

[TABLE]

since $\|P\|_{\mathbf{X}_{\theta}\rightarrow\mathbf{Y}_{\theta}}\leq 1$. Hence, keeping in mind that $b_{n}=\sup_{k\in\mathbb{Z}}\|\widehat{P\varphi}_{n}(k)\|_{\mathbf{Y}_{\theta}}\xrightarrow[n\rightarrow\infty]{\,}0$, we can conclude from (3.1) that

[TABLE]

for every $y^{*}\in B_{Y_{0}^{*}}$.

Next, if we call $\psi_{n}(t)=\sum_{k\notin A_{n}(N_{n})}\hat{\varphi}_{n}(k)$, we can deduce that

[TABLE]

To justify this last equality, we will give the details for the case where the degree of the polynomial is $2$. The reader shall keep in mind that the general case follows the same steps, with the appropriate adaptation.

[TABLE]

Apply finally Lemmas 2.6 and 2.7 to conclude the proof.

∎

Corollary 3.2**.**

For $\mathbf{X},\,\mathbf{Y}$ and $P$ as in Theorem 3.1 and for $\left<X_{0},X_{1}\right>_{\theta}$, the Banach space that appears when applying Peetre’s Interpolation method, we have that $P:\left<X_{0},X_{1}\right>_{\theta}\rightarrow\mathbf{Y}_{\theta}$ is compact for every $0<\theta<1$.

Proof.

First of all, notice that $\left<X_{0},X_{1}\right>_{\theta}$ is contained in $\mathbf{X}_{\theta}$ (as pointed out in [21, 28]) and hence we may use all the previous results. More specifically, observe that if $x\in\left<X_{0},X_{1}\right>_{\theta}$, then we can write $x=\sum_{k\in\mathbb{Z}}x_{k}$, with the series converging unconditionally and therefore

[TABLE]

for $j=0,\,1$. Hence, $\varphi(z):=\sum_{k=1}^{\infty}e^{-\theta k}x_{k}z^{k}$ is holomorphic on $1<|z|<e$ and continuous on the boundary (because the series converges uniformly), therefore is an element of $\mathcal{F}(\mathbf{X})$.

Therefore, $B_{\left<X_{0},X_{1}\right>_{\theta}}\subseteq\mho_{\theta}$ and result follows.

∎

Keeping in mind that compactness of a holomorphic function $f$ and compactness of each of the polynomials that appear in the Taylor series representation of $f$ are equivalent ([2]), we also have the following corollary:

Corollary 3.3**.**

*Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be compatible couples of Banach spaces and let $f:(X_{0},X_{1})\rightarrow(Y_{0},Y_{1})$ so that $f:X_{0}+X_{1}\rightarrow Y_{0}+Y_{1}$ and $f:X_{j}\rightarrow Y_{j}$ ($j=0,1$) are holomorphic. Assume furthermore that $f|_{X_{0}}:X_{0}\rightarrow Y_{0}$ is compact.

Then, $f:\left<X_{0},X_{1}\right>_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is compact, for every $0<\theta<1$.*

Proof.

We remark first that, by means of Lemma 2.3, we obtain that $f:[X_{0},X_{1}]_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is holomorphic, so, applying once more the fact that $\left<X_{0},X_{1}\right>_{\theta}$ is contained in $[X_{0},X_{1}]_{\theta}$, we deduce that $f:\left<X_{0},X_{1}\right>\rightarrow[Y_{0},Y_{1}]_{\theta}$ is holomorphic as well.

Furthermore, a look at the details in the proof of Lemma 2.3 shows that the sequence of polynomials that gives holomorphy at one point $x$ is $\left\{{d^{m}f(x)\over m!}\right\}_{m=0}^{\infty}$. Applying corollary 3.3, we obtain that

[TABLE]

is compact for every $m\geq 0$, and therefore $f:\left<X_{0},X_{1}\right>_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is compact as well. ∎

4. Generalization of classic interpolation results.

Theorem 4.1**.**

*Let $\mathbf{X}=(X_{0},X_{1})$ and $\mathbf{Y}=(Y_{0},Y_{1})$ be compatible couples of Banach spaces and assume that we can find $\{y_{n}\}_{n=1}^{\infty}\subseteq Y_{0}\cap Y_{1}$ so that $\{y_{n}\}_{n=1}^{\infty}$ is a Schauder basis of $(Y_{0}\cap Y_{1},\|\cdot\|_{Y_{j}})$ for both $j=0$ and $j=1$. Assume furthermore that $P\in\mathcal{P}(^{k}\mathbf{X};\mathbf{Y})$ is a continuous homogeneous polynomial for some $k\in\mathbb{N}$ and that $P:X_{0}\rightarrow Y_{0}$ is compact.

Then, for every $0\leq\theta\leq 1$, $P:[X_{0},X_{1}]_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is compact.*

Proof.

First of all, notice that we may assume that $Y_{0}\cap Y_{1}$ is dense in $Y_{0}$ and $Y_{1}$. Indeed, otherwise we may center our attention on $P:([X_{0},X_{1}]_{0},[X_{0},X_{1}]_{1})\rightarrow([Y_{0},Y_{1}]_{0},[Y_{0},Y_{1}]_{1})$, which (as before) is well defined since $P$ is a homogeneous continuous polynomial and, therefore, $P\circ\varphi\in\mathcal{F}\{Y_{0},Y_{1}\}$ for every $\varphi\in\mathcal{F}\{X_{0},X_{1}\}$.

Using Theorem 1.9, we would have that indeed $Y_{0}\cap Y_{1}$ is dense in both $[Y_{0},Y_{1}]_{0}$ and $[Y_{0},Y_{1}]_{1}$ and that [Y_{0},Y_{1}]_{\theta}=\Big{[}[Y_{0},Y_{1}]_{0},[Y_{0},Y_{1}]_{1}\Big{]}_{\theta} (so we would be dealing with the same interpolated spaces). Notice that $[Y_{0},Y_{1}]_{0}$ is a closed subspace of $Y_{0}$ and that the norm in $[Y_{0},Y_{1}]_{0}$ is the same as $\|\cdot\|_{Y_{0}}$ and that we would still have that $P:[X_{0},X_{1}]_{0}\rightarrow[Y_{0},Y_{1}]_{0}$ is compact. Define, for every $n\in\mathbb{N}$, $\pi_{n}:Y_{0}\cap Y_{1}\rightarrow Y_{0}\cap Y_{1}$ as $\pi_{n}(\sum_{i=1}^{\infty}a_{i}y_{i})=\sum_{i=1}^{n}a_{i}y_{i}$. Then, we know the following:

• •

$\pi_{n}$ is a finite rank operator and therefore it is compact. Due to the fact that $\{y_{i}\}_{i=1}^{\infty}$ is a Schauder basis, we deduce that $\pi_{n}$ is continuous.

• •

For every $y\in Y_{0}\cap Y_{1}$ and $j=0,1$, we have $\|y-\pi_{n}y\|_{Y_{j}}\xrightarrow[n\rightarrow\infty]{\,}0$. In particular, $\{y_{n}\}_{n=1}^{\infty}$ is also a Schauder basis for $(Y_{0}\cap Y_{1},\|\cdot\|_{Y_{0}\cap Y_{1}})$.

• •

There exists, for $j=0,1$, $K_{j}\geq 1$ so that $\|\pi_{n}\|_{j}\leq K_{j}$ for every $n\in\mathbb{N}$.

We remark that $\pi_{n}$ admits a continuous extension to $Y_{0}$ and $Y_{1}$, maintaining the norm, and therefore it is possible to extend such operators to the whole $Y_{0}+Y_{1}$ via $\pi_{n}(x_{0}+x_{1})=\pi_{n}(x_{0})+\pi_{n}(x_{1})$. In particular, via the Riesz-Thorin theorem, $\pi_{n}$ is a bounded operator when defined over $[Y_{0},Y_{1}]_{\theta}$. Let us stress that they are still compact operators since they are of finite range.

Let us show that $\pi_{n}P\xrightarrow[n\rightarrow\infty]{\|\cdot\|_{\theta}}P$, so then $P$ would be the limit of compact operators, and hence compact. Indeed, notice first

[TABLE]

Let now $\varepsilon>0$. By compactness, we know that we can find $x_{1},\ldots,x_{m}\in\overline{B}_{X_{0}}$ so that

[TABLE]

We can also find $n_{\varepsilon}\in\mathbb{N}$ so that, for every $n\geq n_{\varepsilon}$ and $j=1,\ldots,m$,

[TABLE]

Therefore, if $n\geq n_{\varepsilon}$ and $x\in\overline{B}_{X_{0}}$, we can choose $1\leq j_{0}\leq m$ so that

[TABLE]

and set

[TABLE]

Hence,

[TABLE]

and we can then conclude, for every $n\geq n_{\varepsilon}$, that

[TABLE]

∎

Corollary 4.2**.**

*Let $(X_{0},X_{1})$ be a compatible couple of Banach spaces and $Y_{0},\,Y_{1}$ be either $L^{q_{0}}(K)$ or $L^{q_{1}}(K)$ (with $K\subseteq\mathbb{R}^{n}$ a compact set and $1\leq q_{0},\,q_{1}\leq\infty$) or $\ell^{q_{0}},\,\ell^{q_{1}}$ ($1\leq q_{0},\,q_{1}<\infty$). Assume $k\in\mathbb{N}$ and $P\in\mathcal{P}(^{k}(X_{0},X_{1}),(Y_{0},Y_{1}))$ so that $P:X_{0}\rightarrow Y_{0}$ is compact.

Then, $P:[X_{0},X_{1}]_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is compact for every $0\leq\theta\leq 1$.*

Proof.

Just notice that $[Y_{0},Y_{1}]_{\theta}\in\{L^{q}(K),\,\ell^{q}\}$ with ${1\over q}={1-\theta\over q_{0}}+{\theta\over q_{1}}$ and those spaces admit a Schauder basis which is common for all of the interpolated spaces (the dual to the coordinate operators, $\{e_{n}\}_{n=1}^{\infty}$, for the $\ell^{p}$ spaces and the Haar system for the $L^{p}(K)$ spaces). ∎

In 1957, Pełczynski showed that if $P:\ell_{p}\rightarrow\ell_{q}$ is a bounded homogeneous polynomial of degree $n$, then $P$ is compact, provided $nq<p$ ([29]). Keeping that in mind, we have the following result:

Corollary 4.3**.**

Let $1\leq p,q<\infty$ and $P:\ell_{p}\rightarrow\ell_{q}$ be a homogeneous bounded polynomial of degree $n$ which is not compact. Then, if there exists $1\leq r\leq p$ and $\varepsilon>0$ so that $P(\ell_{r})\subseteq\ell_{{r\over n}-\varepsilon}$, then $P:(\ell_{r},\|\cdot\|_{r})\rightarrow(\ell_{{r\over n}-\varepsilon},\|\cdot\|_{{r\over n}-\varepsilon})$ is not bounded.

Proof.

Indeed, otherwise we would have that $P:(\ell_{r},\|\cdot\|_{r})\rightarrow(\ell_{{r\over n}-\varepsilon},\|\cdot\|_{{r\over n}-\varepsilon})$ is compact, applying the result by Pełczynski. If we consider now

[TABLE]

we would have then that, for every $0\leq\theta\leq 1$, $P:[\ell_{r},\ell_{p}]_{\theta}\rightarrow[\ell_{q},\ell_{{r\over n}-\varepsilon}]_{\theta}$ is compact. In particular, taking $\theta=0$, we have that $[\ell_{p},\ell_{r}]_{0}=\ell_{p}$ and $[\ell_{q},\ell_{{r\over n}-\varepsilon}]_{0}=\ell_{q}$, so $P:\ell_{p}\rightarrow\ell_{q}$ would be compact, reaching hence a contradiction. ∎

The following theorem generalizes a result presented by Lions and Peetre in [24] (Theorems 2.1 and 2.2 from ch. V, pp. 36–37).

Theorem 4.4**.**

Let $(X_{0},X_{1})$ and $(Y_{0},Y_{1})$ be two compatible couples of Banach spaces, $X,\,Y$ be Banach spaces and let $0<\theta<1$.

1. (1)

If $P\in\mathcal{P}(^{m}X,(Y_{0},Y_{1}))$ and $P:X\rightarrow Y_{0}$ is compact, then $P:X\rightarrow\mathbf{Y}_{\theta}$ is compact. 2. (2)

If $P\in\mathcal{P}(^{m}(X_{0},X_{1}),Y)$ and $P:X_{0}\rightarrow Y$ is compact, then $P:\mathbf{X}_{\theta}\rightarrow Y$ is compact.

Before giving the details of the proof, we will need to state the following technical lemma:

Lemma 4.5**.**

Let $X,\,Y$ be two Banach spaces and let $T:\stackrel{{\scriptstyle m}}{{\overbrace{X\times\ldots\times X}}}\rightarrow Y$ be a symmetric multilinear operator. Then, for every $x_{0},x_{1}\in X$, we have

[TABLE]

Proof.

We will proceed via induction on $m$. For $m=1$, the result is trivial, since then the claim is just the linearity of $T$.

Assume the result is true for $m$. Then,

[TABLE]

∎

Proof of Theorem 4.4.

hola

1. (1)

Let $\{x_{n}\}_{n=1}^{\infty}\subseteq X$ be a bounded sequence. Then, we can find a subsequence $\{x_{n_{k}}\}_{k=1}^{\infty}$ so that $\{Px_{n_{k}}\}_{k=1}^{\infty}\subseteq Y_{0}\cap Y_{1}$ is a Cauchy sequence with respect to $\|\cdot\|_{Y_{0}}$. Now, notice that

[TABLE] 2. (2)

Let us show that $P(B_{\mathbf{X}_{\theta}})$ is a relatively compact subset of $Y$. Indeed, let $\varepsilon>0$. Let $t\geq 1$ so that

[TABLE]

where $m$ is the degree of $P$ and $C^{\prime}>0$ is the constant given by Theorem 1.10, part 2. Apply next relative compactness of $P(B_{X_{0}})$ to find elements $x^{(1)},\ldots,x^{(n)}\in B_{X_{0}}$ so that

[TABLE]

Then, if $x\in B_{\mathbf{X}_{\theta}}$, we can apply the Theorem 1.10, part 2, to obtain a decomposition $x=x_{0}+x_{1}$ with $x_{0}\in X_{0}$, $x_{1}\in X_{1}$, $\|x_{0}\|_{X_{0}}\leq C^{\prime}t^{\theta}$ and $\|x_{1}\|_{X_{1}}\leq C^{\prime}t^{\theta-1}$. Choose also $1\leq j_{0}\leq n$ so that $\|P(x_{0})-P(x^{(j_{0})})\|_{Y}<{\varepsilon\over 2}$. Then, using Lemma 4.5

[TABLE]

Therefore, we can conclude that $P(B_{[X_{0},X_{1}]_{\theta}})\subseteq\cup_{j=1}^{n}B(P(x^{(j_{0})});\varepsilon)$.

∎

Theorem 4.6**.**

*Let $\mathbf{X}=(X_{0},X_{1}),\,\mathbf{Y}=(Y_{0},Y_{1})$ be two compatible couples of Banach spaces and let $P\in\mathcal{P}(^{n}\mathbf{X};\mathbf{Y})$ be a bounded homogeneous polynomial, so that $P_{|X_{0}}:X_{0}\rightarrow Y_{0}$ is compact. Assume that we can find a family of polynomials $\{P_{\lambda}:Y_{0}+Y_{1}\rightarrow Y_{0}\cap Y_{1}\}_{\lambda\in\Lambda}$ and a constant $C>0$ so that $\|P_{\lambda}\|_{Y_{j},Y_{j}}\leq C$ (for $j=0,1$ and $\lambda\in\Lambda$) and, for every $\varepsilon>0$ we can find $\lambda_{\varepsilon}\in\Lambda$ so that $\|Px-P_{\lambda_{\varepsilon}}x\|_{Y_{0}}<\varepsilon$ for every $x\in B_{[X_{0},X_{1}]_{\theta}}$.

Then, $P:[X_{0},X_{1}]_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is compact.*

Proof.

Given $\lambda\in\Lambda$, define the homogeneous polynomial $Q_{\lambda}=P_{\lambda}\circ P:X_{0}+X_{1}\rightarrow Y_{0}\cap Y_{1}\hookrightarrow[Y_{0},Y_{1}]_{\theta}$. Then, in particular, $Q_{\lambda}:X_{0}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is compact and, applying Theorem 4.4, part 2, we get that $Q_{\lambda}:[X_{0},X_{1}]_{\theta}\rightarrow[Y_{0},Y_{1}]_{\theta}$ is compact.

Let us show that we can approximate $P$ by $\{Q_{\lambda}\}_{\lambda\in\Lambda}$, in the uniform norm. Indeed, let $\varepsilon>0$ and let $C>0$ be the constant given by the hypothesis. We can then find $\lambda_{0}\in\Lambda$ so that

[TABLE]

for every $x\in B_{X_{0}}$. Then, using Theorem 1.8

[TABLE]

∎

Acknowledgements The present paper was completed while the author was completing his Ph.D. in Kent State University. The author would also like to thank Professor M. Cwikel for his selfless help in filling the details of his results in [14]. The author is specially grateful to Professor Richard M. Aron for proposing this topic to him and his advice and guidance throughout its contents.