Interpolation of compact bilinear operators

TL;DR

This paper studies how the compactness of bilinear operators behaves under interpolation of Banach spaces, providing a general framework and new theorems applicable to various interpolation methods.

Contribution

It introduces a unified approach for analyzing the stability of compactness in bilinear operators across different interpolation schemes.

Findings

Established a one-sided bilinear interpolation theorem for compactness.

Applied the framework to Peetre's and real interpolation methods.

Demonstrated the stability of compactness under these interpolation techniques.

Abstract

We investigate the stability of compactness of bilinear operators acting on the product of interpolation of Banach spaces. We develop a general framework for such results and our method applies to abstract methods of interpolation in the senseof Aronszajn and Gagliardo. A key step is to show an one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an approximation property. We show applications to general cases, including Peetre's method and the general real interpolation methods.

Equations

$$\Vert T(x,y){\Vert }_{Z_j}\le M_j\Vert x{\Vert }_{X_j}\Vert y{\Vert }_{Y_j},(x,y)\in (X_0\cap X_1)\times (Y_0\cap Y_1).$$

$$\|T\|_{X\times Y\to Z}:=\sup\big{\{}\|T(x,y)\|_{Z};\,(x,y)\in B_{X\times Y}\big{\}}.$$

$$(S_0,S_1)(x_0,x_1):=(S_0{x}_0,S_1{x}_1),(x_0,x_1)\in X_0\times X_1.$$

$$K(s,t,x;\vec{X})=\inf \{s\Vert x_0{\Vert }_{X_0}+t\Vert x_1{\Vert }_{X_1};x=x_0+x_1\},x\in X_0+X_1.$$

$$K(t,x;\vec{X}):=K(1,t,x;\vec{X}),t>0.$$

$${\varphi }_X(s,t)=\sup \{\Vert x{\Vert }_X;x\in X_0\cap X_1,\Vert x{\Vert }_{X_0}\le s,\Vert x{\Vert }_{X_1}\le t\}$$

$${\psi }_X(s,t;\vec{X}):=\sup \{K(s,t,x;\vec{X});\Vert x{\Vert }_X=1\},s,t>0).$$

$$\|T\|_{X\times Y\to Z}\leq\varphi\big{(}\|T\|_{X_{0}\times Y_{0}\to Z_{0}},\|T\|_{X_{1}\times Y_{1}\to Z_{1}}\big{)},$$

$$\underset{n\to \infty }{\lim }\Vert T_n(a_n,b_n){\Vert }_C=\lambda .$$

$$\Vert T(a_n,b_n)-T(a_k,b_k){\Vert }_C$$

$$\displaystyle\leq 2\phi_{C}\big{(}\|T(a_{n},b_{n})-T(a_{k},b_{k})\|_{C_{0}},1\big{)}.$$

$$\Vert T{\Vert }_{(A_0+A_1)\times (B_0+B_1)\to C}\le 1,\underset{j=0,1}{\max }\Vert T{\Vert }_{A_j\times B_j\to C}\le 1.$$

$$\underset{t\to \infty }{\lim }\frac{{\psi }_0(t)}{t}=0,\underset{t\to \infty }{\lim }\frac{{\psi }_1(t)}{t}=0.$$

$$\max\bigg{\{}\frac{\psi_{0}(t)}{t},\,\frac{\psi_{1}(t)}{t},\,\frac{\psi_{0}(t)}{t}\,\frac{\psi_{1}(t)}{t}\bigg{\}}<\frac{\varepsilon}{8}.$$

$$K(t,a_n;\vec{A})\le {\psi }_0(t),K(t,b_n;\vec{B})\le {\psi }_1(t).$$

$$\Vert a_n^0{\Vert }_{A_0}+t\Vert a_n^1{\Vert }_{A_1}\le 2K(t,a_n;\vec{A}),\Vert b_n^0{\Vert }_{B_0}+t\Vert b_n^1{\Vert }_{B_1}\le 2K(t,a_n;\vec{A}).$$

$$\Vert a_n^0{\Vert }_{A_0}+t\Vert a_n^1{\Vert }_{A_1}\le 2{\psi }_0(t),\Vert b_n^0{\Vert }_{B_0}+t\Vert b_n^1{\Vert }_{B_1}\le 2{\psi }_1(t).$$

$$\Vert T(a_n^0,b_n^0)-z{\Vert }_C<\frac{\epsilon }{4},n>N.$$

$$T(a_n,b_n)=T(a_0^n,b_0^n)+T(a_0^n,b_1^n)+T(a_1^n,b_n),n\in \mathbb{N}.$$

$$\Vert T(a_n,b_n)-z{\Vert }_C$$

$$\le \frac{\epsilon }{4}+\Vert T(a_n,b_1^n){\Vert }_C+\Vert T(a_1^n,b_1^n){\Vert }_C+\Vert T(a_1^n,b_n){\Vert }_C$$

$$\le \frac{\epsilon }{4}+\Vert a_n{\Vert }_X\Vert b_1^n{\Vert }_C+\Vert a_1^n{\Vert }_{A_1}\Vert b_1^n{\Vert }_{B_1}+\Vert a_1^n{\Vert }_{A_1}\Vert b_n{\Vert }_C$$

$$\le \frac{\epsilon }{4}+2\frac{{\psi }_1(t)}{t}+4\frac{{\psi }_0(t)}{t}\frac{{\psi }_1(t)}{t}+2\frac{{\psi }_1(t)}{t}\le \epsilon .$$

$$C:=\sup_{n\in\mathbb{N}}\big{\{}\|P_{n}\|_{\vec{A}\to\vec{A}}\,,\|P_{n}^{+}\|_{\vec{A}\to\vec{A}}\,,\|P_{n}^{-}\|_{\vec{A}\to\vec{A}}\big{\}}<\infty.$$

$$I=P_n+P_n^{+}+P_n^{-},n\in \mathbb{N}.$$

$$\underset{n\to \infty }{\lim }\Vert P_n^{+}{\Vert }_{A_0\to A_1}=\underset{n\to \infty }{\lim }\Vert P_n^{-}{\Vert }_{A_1\to A_0}=0.$$

$$T$$

$$+T(P_n^{+},Q_n)+T(P_n^{+},Q_n^{+})+T(P_n^{+},Q_n^{-})+T(P_n^{-},Q_n)+T(P_n^{-},Q_n^{+})+T(P_n^{-},Q_n^{-})$$

$$=T(P_n,Q_n)+T(P_n,Q_n^{+})+(R_n+R_n^{+}+R_n^{-})T(P_n,Q_n^{-})+T(P_n^{+},Q_n)+T(P_n^{+},Q_n^{+})$$

$$+T(P_n^{+},Q_n^{-})+T(P_n^{-},Q_n)+T(P_n^{-},Q_n^{+})+(R_n+R_n^{+}+R_n^{-})T(P_n^{-},Q_n^{-})$$

$$=T(P_n,Q_n)+T(P_n,Q_n^{+})+R_{n}T(P_n,Q_n^{-})+R_n^{+}T(P_n,Q_n^{-})+R_n^{-}T(P_n,Q_n^{-})$$

$$+T(P_n^{+},Q_n)+T(P_n^{+},Q_n^{+})+T(P_n^{+},Q_n^{-})+T(P_n^{-},Q_n)+T(P_n^{-},Q_n^{+})$$

$$+R_{n}T(P_n^{-},Q_n^{-})+R_n^{+}T(P_n^{-},Q_n^{-})+R_n^{-}T(P_n^{-},Q_n^{-}).$$

$$T(P_n,Q_n):A\times B\stackrel{(P_n,Q_n)}{⟶}(A_0\cap A_1)\times (B_0\cap B_1)\hookrightarrow A_j\times B_j\stackrel{T}{⟶}{C}_j,$$

$$R_{n}T(P_n,Q_n^{-}):A_j\times B_j\stackrel{T(P_n,Q_n^{-})}{⟶}{C}_j\stackrel{R_n}{⟶}{C}_0\cap C_1\hookrightarrow C,$$

Full text

††footnotetext: 2010 Mathematics Subject Classification: Primary 46B70, 47B07.††footnotetext: Key words and phrases: Compact bilinear operators, interpolation spaces, interpolation functors.††footnotetext: The first named author was supported by the National Science Centre of Poland, project 2015/17/B/ST1/00064.

1 Introduction

In recent years various properties of bilinear and as multilinear operators are being studied intensively. Interest in this study has increased since these operators are connected to important applications. We mention applications in harmonic analysis in the study of $p$-Sidon sets (see [3]). Bilinear operators appear in applications in elasticity. We point out the Newton–Kantorovič effective method for solving certain equations involving bilinear operators on Banach spaces (see [1]). These operators also play an important role in scattering theory (see [16]). The bilinear interpolation theorems are powerful tools in the theory of Banach operator ideals.

An important question related to the behavior of interpolation of compact operators is whether an operator acting between Banach couples and compactly on one or both of the ‘endpoint’ spaces, also acts compactly on the interpolation spaces generated by the couples. It is a natural question if there are variants of known results in the setting of bilinear operators. In current paper, we discuss interpolation of bilinear compact operators. The problem of interpolation of bilinear operators by the classical real method was first studied by Lions and Peetre in their seminal paper [17]. Calderón studied the same problem in his fundamental paper [8] for the lower complex method. In addition the interpolation of compact bilinear operators is also considered in [8, 10.4]. The counterpart has been studied recently in [14] for the real method $(\,\cdot\,)_{\rho,q}$ with quasi-power function parameter $\rho$ and $1\leq q\leq\infty$, which is a generalization of the classical real method generated by $\rho(t)=t^{\theta}$ for all $t>0$ with $\theta\in(0,1)$. Results from [14] were extended in [12] for larger class of real methods of interpolation.

The problem of interpolation of bilinear operators by abstract interpolation methods was studied in [18, 19]. The stability of compactness of bilinear operators acting on the product of the real interpolation spaces has been studied recently as well as in [9, 13, 20]. We also mention that in a very recent paper [6] the authors established an interesting formula for the measure of non-compactness of bilinear operators interpolated by the general real method. In particular this result applies to the real method and to the real method with a function parameter.

The investigation on compactness property of bilinear operators acting on the product of abstract interpolation Banach spaces is not currently much advanced. In a recent paper [20], interpolation of the measure of non-compactness of bilinear operators is studied. In this paper the results of a general nature are proved which states that, for a large class of interpolation functors preserving bilinear interpolation estimates of measures of non-compactness of interpolated linear operators between Banach couples, can be lifted to bilinear operators. It has been shown that, as an application, the measure of non-compactness of bilinear operators behave well under the real method of interpolation. Applications of these results comprise theorems on stability of compactness of interpolated operators.

We point out that these results are proved for the class of bilinear operators $T$ defined on the products of intersections $(X_{0}\cap X_{1})\times(Y_{0}\cap Y_{1})$ of Banach couples $(X_{0},X_{1})$ and $(Y_{0},Y_{1})$ with values for the intersection $Z_{0}\cap Z_{1}$ of a Banach couple $(Z_{0},Z_{1})$, such that, for both $j=0$ and $j=1$, we have

[TABLE]

The study of abstract interpolation properties of this class of bilinear operators requires some natural restrictions whenever we expect to prove an abstract general result. It should be pointed out that many important bilinear operators in harmonic analysis belong to the above type defined for a special class of Banach function spaces. We refer to [4] and [6], where compactness of commutators of bilinear Calderón–Zygmund operators and multiplication by functions in $C\!M\!O$ of $B\!M\!O$ from the product $L_{p}\times L_{q}$ into $L_{r}$ is studied under the conditions $1<p,q<\infty$ and $1/p+1/q=1/r\leq 1$.

In this paper, we provide a very general abstract approach in the study of the stability of compactness property of (bounded) bilinear operators acting on products of abstract interpolation of Banach spaces. We consider bilinear operators $T\colon(X_{0}+X_{1})\times(Y_{0}+Y_{1})\to Z_{0}+Z_{1}$, such that the restriction $T\colon X_{j}\times Y_{j}\to Z_{j}$ is bounded for $j=0$ and $j=1$. We prove an one-sided bilinear interpolation theorem on compactness for bilinear operators of this type, acting on couples satisfying an approximation property $(Ap)$, introduced in a remarkable paper by Cobos and Peetre [11]. Result is lifted to the wider class of abstract methods of interpolation in the sense of Aronszajn and Gagliardo, allowing us to obtain a very general compactness result for interpolation of bilinear operators. As applications, we consider the real, complex and Peetre interpolation methods.

2 Definitions and preliminary results

We use notations from Banach space theory. The (closed) unit ball of a Banach space $X$ is denoted by $B_{X}$. As usual, we denote by $\mathcal{L}(X,Y)$, the Banach space of all bounded operators $T\colon X\to Y$ from Banach space $X$ into $Y$, equipped with uniform norm.

The product $X\times Y$ of two Banach spaces is equipped with the norm $\|(x,y)\|_{X\times Y}:=\max\{\|x\|_{X},\|y\|_{Y}\}$ for all $(x,y)\in X\times Y$. $\mathcal{L}_{2}(X\times Y,Z)$ denotes the Banach space of all $2$-linear bounded mappings $T\colon X\times Y\to Z$, equipped with the norm

[TABLE]

Mapping $T\in\mathcal{L}_{2}(X\times Y,Z)$ is called a bilinear operator.

A $2$-linear mapping $X\times Y\to Z$ is said to be compact if $T$ maps bounded subsets of $X\times Y$ into precompact subsets of $Z$. This condition is equivalent to precompactness of $T(B_{X\times Y})$ in $Z$. We will use an equivalent condition, namely for any bounded sequence $\{(x_{n},y_{n})\}$ in $X\times Y$, the sequence $\{T(x_{n},y_{n})\}$ has a convergent subsequence in $Z$. We refer to [4] for examples of bilinear compact operators.

If $S_{0}\colon X_{0}\to Y_{0}$ and $S_{1}\colon X_{1}\to Y_{1}$ are operators between Banach spaces, then we denote by $(S_{0},S_{1})$ the bounded linear operator from $X_{0}\times X_{1}$ to $Y_{0}\times Y_{1}$ defined by

[TABLE]

The following obvious proposition is required.

Proposition 2.1**.**

Let $S_{0}\colon X_{0}\to Y_{0}$ and $S_{1}\colon X_{1}\to Y_{1}$ be surjective operators between Banach spaces. Suppose that $W$ and $Z$ are Banach spaces and let $T\colon Y_{0}\times Y_{1}\to Z$ be a bilinear operator. If $V\colon Z\to W$ is an isomorphic embedding, then $T\colon Y_{0}\times Y_{1}\to Z$ is compact if, and only if, the bilinear operator $VT(S_{0},S_{1})\colon X_{0}\times X_{1}\to W$ is compact.

We will use standard notation from the interpolation theory. As a rule, we follow [5]. If $X$ is an intermediate Banach space with respect to a couple $\vec{X}=(X_{0},X_{1})$, we let $X^{\circ}$ be the closed hull of $X_{0}\cap X_{1}$ in $X$, and the Banach couple $(X_{0}^{\circ},X_{1}^{\circ})$ is denoted by $\vec{X}^{\circ}$. A Banach couple $(X_{0},X_{1})$ is called regular if $X_{j}^{\circ}=X_{j}$ for $j\in\{0,1\}$.

We shall recall that a mapping $\mathcal{F}$ from the category of all couples of Banach spaces into the category of all Banach spaces is said to be an interpolation functor (or method) if, for any couple $\vec{X}:=(X_{0},X_{1})$, the Banach space $\mathcal{F}(X_{0},X_{1})$ is intermediate with respect to $\vec{X}$ (i.e., $X_{0}\cap X_{1}\hookrightarrow\mathcal{F}(\vec{X})\hookrightarrow X_{0}+X_{1}$), and $T\colon\mathcal{F}(X_{0},X_{1})\to\mathcal{F}(Y_{0},Y_{1})$ for all $T\colon(X_{0},X_{1})\to(Y_{0},Y_{1})$ As usual, the notation $T\colon(X_{0},X_{1})\to(Y_{0},Y_{1})$ means that $T\colon X_{0}+X_{1}\to Y_{0}+Y_{1}$ is a linear operator such that the restrictions of $T$ to space $X_{j}$ is a bounded operator from $X_{j}$ to $Y_{j}$, for both $j=0$ and $j=1$. The interpolation functor is said to be exact if $\|T\|_{\mathcal{F}(X_{0},X_{1})\to\mathcal{F}(Y_{0},Y_{1})}\leq\max_{j=0,1}\|T|_{X_{j}}\|_{X_{j}\to Y_{j}}.$

The set of all functions $\varphi\colon(0,\infty)\times(0,\infty)\to(0,\infty)$, which are non-decreasing in each variable and positively homogeneous (that is, $\varphi(\lambda s,\lambda t)=\lambda\varphi(s,t)$ for all $\lambda,s,t>0$), is denoted by $\Phi$. The subset of all $\varphi\in\Phi$, such that $\lim_{s\to 0}\varphi(1,s)=\lim_{s\to 0}\varphi(s,1)=0$ is denoted by $\Phi_{0}$.

Note that for any $\varphi\in\Phi$, $\varphi\neq 0$, the function $(s,t)\mapsto 1/\varphi(1/s,1/t)$ defined for all $s,t>0$ also belongs to $\Phi$. This function will be denoted by $\varphi^{*}$. Observe that functions from $\Phi$ are continuous by monotonicity. Note that every $\varphi\in\Phi$ can be extended by continuity to $[0,\infty)\times[0,\infty)$. This extension will be denoted by the same symbol $\varphi$. The simplest examples of interpolation functions are $as+bt$, $\max\{as,bt\}$, and $\min\{as,bt\}$, where $a,b>0$ and the power functions $s^{1-\theta}t^{\theta}$, where $0\leq\theta\leq 1$.

Let $\vec{X}=(X_{0},X_{1})$ be a Banach couple. For every $s,t>0$, we define the $K$-functional

[TABLE]

In the sequel, for $x\in X_{0}+X_{1}$,

[TABLE]

For any Banach space $X$, such that $X_{0}\cap X_{1}\hookrightarrow X$ (resp., $X\hookrightarrow X_{0}+X_{1}$), we define (the fundamental function of $X$ with respect to $\vec{X}$) $\phi_{X}\in\Phi$ (resp., $\psi_{X}\in\Phi$) by

[TABLE]

(resp.,

[TABLE]

Let $\vec{X}=(X_{0},X_{1})$, $\vec{Y}=(Y_{0},Y_{1})$ and $\vec{Z}=(Z_{0},Z_{1})$ be Banach couples. If an operator $T\colon(X_{0}+X_{1})\times(Y_{0}+Y_{1})\to Z_{0}+Z_{1}$ is such that the restrictions $T\colon X_{0}\times Y_{0}\to Z_{0}$ and $T\colon X_{1}\times Y_{1}\to Z_{1}$ are bilinear operators, then we write $T\colon\vec{X}\times\vec{Y}\to\vec{Z}$.

Let’s assume that $X$, $Y$ and $Z$ are Banach spaces intermediate to Banach couples $\vec{X}$, $\vec{Y}$ and $\vec{Z}$, respectively. If for every bilinear operator $T\colon\vec{X}\times\vec{Y}\to\vec{Z}$, the restriction of $T$ is bounded from $X\times Y$ to $Z$, then $X$, $Y$ and $Z$ are called bilinear interpolation spaces with respect to $(\vec{X},\vec{Y})$ and $\vec{Z}$ ($(X,Y;Z)\in\mathcal{B}(\vec{X},\vec{Y};\vec{Z})$ for short). If in addition there exists a function $\varphi\in\Phi$, such that

[TABLE]

then $X$, $Y$ and $Z$ are called $\varphi$-bilinear interpolation spaces, and we write $(X,Y;Z)\in\mathcal{B}_{\varphi}(\vec{X},\vec{Y};\vec{Z})$ for short.

The following observation is required.

Proposition 2.2**.**

Let $A,B$ and $C$ be Banach spaces and let $\{T_{n}\}_{n=1}^{\infty}$ be a sequence of bilinear operators from $A\times B$ to $C$, such that $\|T_{n}\|_{A\times B\to C}\to\lambda$ as $n\to\infty$. Then, there exists a sequence $\{(a_{n},b_{n})\}_{n=1}^{\infty}$ in the unit ball of $A\times B$, such that

[TABLE]

We also quote the following technical result. Since the proof is obvious, it will be omitted.

Proposition 2.3**.**

Let $(A_{0},A_{1})$ and $(B_{0},B_{1})$ be Banach couples and let $C$ be a Banach space. Assume that $T\colon(A_{0}+A_{1})\times(B_{0}+B_{1})\to C$ is $2$-linear mapping, such that $T$ is a bilinear operator from $A_{j}\times B_{j}$ to $C$ for $j\in\{0,1\}$. Then $T\colon(A_{0}+A_{1})\times(B_{0}\cap B_{1})\to C$ and $T\colon(A_{0}\cap A_{1})\times(B_{0}+B_{1})\to C$ are bounded bilinear operators.

We will now provide variants of Lions–Peetre compactness results in the setting of bilinear operators.

Lemma 2.4**.**

Let $A$ and $B$ be Banach spaces, $(C_{0},C_{1})$ be a Banach couple and $C$ be a Banach space, such that $C_{0}\cap C_{1}\hookrightarrow C$. Assume that a bilinear operator $T\colon A\times B\to C_{0}\cap C_{1}$ is such that $T\colon A\times B\to C_{0}$ is compact. Then $T\colon A\times B\to C$ is also compact whenever $\phi_{C}(s,1)\to 0$ as $s\to 0$.

Proof.

Without loss of generality we may assume that $\max_{j=0,1}\|T\|_{A\times B\to C_{j}}\leq 1$. Let $\{(a_{n},b_{n})\}$ be a bounded sequence in the unit ball of $A\times B$. Since $T$ is compact from $A\times B$ into $C_{0}$, by passing to subsequence, we may assume that $\{T(a_{n},b_{n})\}$ is a Cauchy sequence in $C_{0}$. Since $\|T(a_{n},b_{n})-T(a_{k},b_{k})\|_{C_{1}}\leq 2$, it follows from monotonicity of function $\phi_{X}$ that for each positive integer $n$ and $k$,

[TABLE]

Combining with our hypothesis that $\phi_{C}(s,1)\to 0$ as $s\to 0+$ yields that $\{T(a_{n},b_{n})\}$ is a Cauchy sequence in $C$. The proof is complete. ∎

The next variant of the Lions–Peetre compactness result for bilinear operators is given in the following lemma.

Lemma 2.5**.**

Let $C$ be any Banach space and $\vec{A}=(A_{0},A_{1})$, $\vec{B}=(B_{0},B_{1})$ be Banach couples and let $T\colon(A_{0}+A_{1})\times(B_{0}+B_{1})\to C$ be a bilinear operator, such that $T\in\mathcal{L}_{2}(A_{j}\times B_{j},C)$ for $j\in\{0,1\}$. Assume that $A\hookrightarrow A_{0}+A_{1}$ and $B\hookrightarrow B_{0}+B_{1}$ are Banach spaces, such that $\psi_{A}(t,1)\to 0$, $\psi_{B}(t,1)\to 0$ as $t\to 0$. Then, for any compact bilinear operator $T\colon A_{0}\times B_{0}\to C$, the restriction $T\colon A\times B\to C$ is also a compact operator.

Proof.

We may assume without loss of generality that the norms of the inclusion maps $A\hookrightarrow A_{0}+A_{1}$ and $B\hookrightarrow B_{0}+B_{1}$ are less than or equal to $1$ and that

[TABLE]

Clearly this implies that $T\colon A\times B_{1}\to Z$ and $T\colon A_{0}\times B\to Z$ are bounded bilinear operators with norms less than or equal to $1$.

To simplify notation, we put $\psi_{0}(t):=\psi_{X}(1,t)$ and $\psi_{1}(t):=\psi_{Y}(1,t)$ for all $t>0$. Our hypothesis about limits is equivalent to

[TABLE]

Let’s fix a sequence $\{(a_{n},b_{n})\}$ in the unit ball of $A\times B$. The assumptions on limits allow us to choose, for a given $\varepsilon>0$, there exists a sufficiently large $t>0$, such that

[TABLE]

From the definition of $\psi_{X}$ and $\psi_{Y}$, it follows that for all $n\in\mathbb{N}$ and chosen $t>0$, we have

[TABLE]

Then, for each $n\in\mathbb{N}$, we find the decompositions $a_{n}=a_{n}^{0}+a_{n}^{1}$ and $b_{n}=b_{n}^{0}+b_{n}^{1}$ with $a_{n}^{j}\in A_{j}$ and $b_{n}^{j}\in B_{j}$ for each $j\in\{0,1\}$, such that

[TABLE]

The combination of these inequalities yields for each $n\in\mathbb{N}$,

[TABLE]

Hence $\{a_{n}^{0}\}$ and $\{b_{n}^{0}\}$ are bounded sequences in $A_{0}$ and $B_{0}$, respectively. Since $T\colon A_{0}\times B_{0}\to C$ is a compact bilinear operator, by passing to a subsequence, if necessary, we may assume that there exists $z\in Z$, such that for some $N=N(\varepsilon)$,

[TABLE]

We claim that $\{T(a_{n},b_{n})\}$ converges to $z$ in $C$. We may observe that

[TABLE]

In combination with the above estimates, for each $n>N$ we have:

[TABLE]

This proves the claim and the proof is complete. ∎

3 Interpolation of compact bilinear operators on couples with approximation property

In this section we prove a key one-sided compactness interpolation theorem for bilinear operators acting on Banach couples which satisfies the approximation property $(Ap)$. Following [11], we recall that a Banach couple $\vec{A}=(A_{0},A_{1})$ satisfies the approximation property $(Ap)$ if there is a sequence $\{P_{n}\}_{n=1}^{\infty}$ of operators from $A_{0}+A_{1}$ into $A_{0}\cap A_{1}$ and two other sequences $\{P^{+}_{n}\}$ and $\{P^{-}_{n}\}_{n=1}^{\infty}$, of operators from $A_{0}+A_{1}$ into $A_{0}+A_{1}$, such that

• (I)

They are uniformly bounded in $\vec{A}$, i.e.,

[TABLE]

• (II)

The identity operator $I$ on $A_{0}+A_{1}$ may be written as

[TABLE]

• (III)

For each $n\in\mathbb{N}$, we have $P^{+}\colon A_{0}\to A_{1}$ and $P^{-}_{n}\colon A_{1}\to A_{0}$, with

[TABLE]

Lemma 3.1**.**

Suppose that a Banach couple $(A_{0},A_{1})$ satisfies the approximation property $(Ap)$. Then the following holds:**

• (i)

If $a\in A_{0}^{\circ}$, then $\|P^{-}_{n}a\|_{A_{0}}\to 0$ as $n\to\infty$;

• (ii)

If $a\in A_{1}^{\circ}$, then $\|P^{+}_{n}a\|_{A_{1}}\to 0$ as $n\to\infty$.

The next theorem is the core for our main result in the following section.

Theorem 3.2**.**

Let $A$, $B$ and $C$ be Banach spaces intermediate to Banach couples $\vec{A}=(A_{0},A_{1})$, $\vec{B}=(B_{0},B_{1})$ and $\vec{C}=(C_{0},C_{1})$, respectively, which satisfy the approximation property $(Ap)$ and $\psi_{A}(s,1)\to 0$, $\psi_{B}(s,1)\to 0$ and $\phi_{C}(s,1)\to 0$ as $s\to 0$. Assume that $(A,B;C)\in\mathcal{B}_{\varphi}(\vec{A},\vec{B};\vec{C})$ and $(A,B;C)\in\mathcal{B}_{\varphi}(\vec{A}^{\circ},\vec{B}^{\circ};\vec{C})$ with $\varphi\in\Phi_{0}$. Then, for any bilinear operator $T\colon(A_{0},A_{1})\times(B_{0},B_{1})\to(C_{0},C_{1})$, such that $T\colon A_{0}\times B_{0}\to C_{0}$ is compact, it follows that $T\colon A\times B\to C$ is also compact.

Proof.

Let $\{P_{n}\}$, $\{P^{+}_{n}\}$, $\{P^{-}_{n}\}$, $\{Q_{n}\}$, $\{Q^{+}_{n}\}$, $\{Q^{-}_{n}\}$ and $\{R_{n}\}$, $\{R^{+}_{n}\}$, $\{R^{-}_{n}\}$ be the corresponding approximating sequences in the Banach couples $\vec{A}$, $\vec{B}$ and $\vec{C}$, respectively, satisfying the approximation property $(Ap)$.

To prove that $T\colon A\times B\rightarrow C$ is compact, we consider the following decomposition:

[TABLE]

We claim that each one of the bilinear operators: $T(P_{n},Q_{n})$, $R_{n}T(P_{n},Q_{n}^{-})$, $R_{n}T(P_{n}^{-},Q_{n}^{-})$, $R_{n}^{+}T(P_{n},Q_{n}^{-})$ and $R_{n}^{+}T(P_{n}^{-},Q_{n}^{-})\}$ are compact from $A\times B$ to $C$, for each $n\in\mathbb{N}$.

Several steps are required.

(i) We start with $T(P_{n},Q_{n})$ by using the following factorization for $j\in\{0,1\}$:

[TABLE]

Since $T\colon A_{0}\times B_{0}\to C_{0}$ is compact, it follows, by Lemma 2.4, that the bilinear operator $T(P_{n},Q_{n})\colon A\times B\to C$ is compact.

(ii) Using the following factorization of $R_{n}T(P_{n},Q^{-}_{n})$, for each $j\in\{0,1\}$:

[TABLE]

we conclude, by Lemma 2.5, that $R_{n}T(P_{n},Q^{-}_{n})\colon A\times B\rightarrow C$ is compact operator.

(iii) Now let us consider the following factorization for $j\in\{0,1\}$,

[TABLE]

Since $T\colon A_{0}\times B_{0}\to C_{0}$, Lemma 2.5 applies. Therefore, $R_{n}T(P_{n}^{-},Q_{n}^{-})$ is a compact operator from $A\times B$ to $C$.

(iv) To show the compactness of $R_{n}^{+}T(P_{n},Q^{-}_{n})$ for each $n$ and since $T\colon A_{0}\times B_{0}\to C_{0}$ is compact, we observe that

[TABLE]

is also compact. Since $R_{n}^{+}\colon(C_{0},C_{0})\to(C_{0},C_{1})$, $R_{n}^{+}\colon C_{0}\to C_{0}\cap C_{1}$ is a bounded operator for each $n\in\mathbb{N}$. Consequently, we conclude that $R_{n}^{+}T(P_{n},Q^{-}_{n})\colon A_{0}\times B_{0}\to C_{0}\cap C_{1}$ is a compact operator compact. Then, by Lemma 2.4, it follows that $R_{n}^{+}T(P_{n},Q^{-}_{n})\colon A\times B\to C$ is also compact.

We show that all sequences of norms of bilinear operators from $A\times B$ to $C$ have limit equal to [math]: $\{\|T(P_{n},Q_{n}^{+})\|\}$, $\{\|R_{n}^{-}T(P_{n},Q_{n}^{-})\|\}$, $\{\|T(P_{n}^{+},Q_{n})\|\}$, $\{\|T(P_{n}^{+},Q_{n}^{+})\|\}$, $\{\|T(P_{n}^{+},Q_{n}^{-})\|\}$, $\{\|R_{n}^{-}T(P_{n},Q_{n}^{-})\|\}$, $\{\|T(P_{n}^{+},Q_{n})\|\}$, $\{\|T(P_{n}^{+},Q_{n}^{+})\|\}$, $\{\|T(P_{n}^{+},Q_{n}^{-})\|\}$, $\{\|T(P_{n}^{-},Q_{n})\|\}$, $\{\|T(P_{n}^{-},Q_{n}^{+})\|\}$, $\{\|R_{n}^{-}T(P_{n}^{-},Q_{n}^{-})\|\}$.

(v) We prove that

[TABLE]

Observe that our hypothesis $(A,B;C)\in\mathcal{B}_{\varphi}(\vec{A},\vec{B};\vec{C})$ yields

[TABLE]

where $C\leq\max\{1,\sup_{n\geq 1}\|T(P_{n},Q^{+}_{n})\|_{A_{1}\times B_{1}\to C_{1}}\}<\infty$. Thus, it is enough to prove that $\|T(P_{n},Q^{+}_{n})\|_{A_{0}\times B_{0}\to C_{0}}\to 0$ as $n\to\infty$. Suppose that this is false. By passing to a subsequence, we may assume, without loss of generality, that for some $\lambda>0$,

[TABLE]

It is clear that the sequence $\{T(P_{n},Q^{+}_{n})\}$ of bilinear operators is uniformly bounded in $A_{0}\times B_{0}$. Thus, Proposition 2.2 shows that, by passing to a subsequence, if necessary, we may assume, without loss of generality, that there exists a sequence $\{(a_{n},b_{n})\}$ in the unit ball of $A_{0}\times B_{0}$, such that $\|T(P_{n},Q^{+}_{n})\|_{A_{0}\times B_{0}\to C_{0}}\to\lambda$ as $n\to\infty$ and

[TABLE]

Since $T\colon A_{0}\times B_{0}\to C_{0}$ is compact, by passing a subsequence, if necessary, we may assume that $\{T(P_{n}a_{n},Q^{+}_{n}b_{n})\}$ converges to some element $b$ in $C_{0}$ with $\|b\|_{C_{0}}=\lambda$. We now observe that we have with $K:=\|T\|_{(A_{0}+A_{1})\times(B_{0}+B_{1})\,\to C_{0}+C_{1}}$

[TABLE]

Finally, note that $\|Q^{+}_{n}\|_{B_{0}\to B_{1}}\to 0$ as $n\to\infty$ implies $T(P_{n}a_{n},Q^{+}_{n}b_{n})\to 0$ in $C_{0}+C_{1}$ as $n\to\infty$. Hence $b=0$ and so $\lambda=0$, which is a contradiction.

(vi) Our next task is to prove that $\|R_{n}^{-}T(P_{n},Q^{-}_{n})\|_{A\times B\to C}\to 0$ as $n\to\infty$. Similarly, we have

[TABLE]

for some constant $\widetilde{K}>0$ independent of $n$. Since the sequences of operators $\{P_{n}\}$, $\{Q_{n}^{-}\}$ and $\{R_{n}^{-}\}$ are uniformly bounded, it is enough to prove that

[TABLE]

Suppose, by a contradiction, that this is not true. Then passing to a subsequence, if necessary, we may assume that for some $\delta>0$

[TABLE]

Applying Proposition 2.2, we conclude that there exists a sequence $\{(a_{n},b_{n})\}$ in the unit ball of $A_{0}\times B_{0}$ with $a_{n}\in A_{0}\cap A_{0}$ and $b_{n}\in B_{0}\cap B_{1}$, such that

[TABLE]

Since the sequence $\{(P_{n}a_{n},Q_{n}^{-}b_{n})\}$ is bounded in $A_{0}\times B_{0}$ and $T\colon A_{0}\times B_{0}\to C_{0}$ is compact by passing, if necessary, to a subsequence, we may assume that the sequence $\{T(P_{n}a_{n},Q_{n}^{-}b_{n})\}$ converges in $C_{0}$ to some $c$. Thus, for $N$ large enough, we have

[TABLE]

Observe that $(a_{n},b_{n})\in(A_{0}\cap A_{1})\times(B_{0}\cap B_{1})$ implies $\{T(P_{n}a_{n},Q_{n}^{-}b_{n})\}\subset C_{0}\cap C_{1}$ for each $n$. Recall that $c=\lim_{n\to\infty}T(P_{n}a_{n},Q_{n}^{-}b_{n})$ in $C_{0}$ and so $c\in C_{0}^{\circ}$. Then, by Lemma 3.1,

[TABLE]

which is a contradiction with the above estimate.

(vii) To prove that

[TABLE]

we use our hypothesis $\varphi\in\Phi_{0}$ and we proceed similarly as in (v).

(viii) For the sequence $\{R_{n}^{+}T(P_{n}^{-},Q_{n}^{-})\}$, we have

[TABLE]

where M:=\max\big{\{}1,\sup_{n\geq 1}\|R_{n}^{+}T(P_{n}^{-},Q^{-}_{n})\|_{A_{0}\times B_{0}\to C_{0}}\big{\}}<\infty. By approximation property $(Ap)$, the sequences of operators $\{P_{n}^{-}\}$, $\{Q_{n}^{-}\}$ and $\{R_{n}^{+}\}$ are uniformly bounded. Thus it is sufficient to show that $\|R_{n}^{+}T(P_{n}^{-},Q_{n}^{-})\|_{A_{1}\times B_{1}\to C_{1}}\to 0$ as $n\to\infty$. Consequently, we combine the factorization

[TABLE]

with the estimate

[TABLE]

to deduce, by the approximation property $(Ap)$, that

[TABLE]

We proceed similarly as in the (v) to obtain $\|R_{n}^{-}T(P_{n}^{-},Q^{-}_{n})\|_{A\times B\to C}\to 0$ as $n\to\infty$. The proof is complete. ∎

4 Bilinear compactness theorem for Aronszajn–Gagliardo functors

In this section we apply our results to bilinear operators acting from the product of interpolation spaces generated by orbit functors to Banach spaces generated by coorbit functors, in the sense of Aronszajn-Gagliardo. We recall two important constructions of the abstract interpolation theory by Aronszajn and Gagliardo [2].

As usual, for non-empty set $\Gamma$ and any Banach space $X$, we denote by $\ell_{1}(\Gamma,X)$ (resp., $\ell_{\infty}(\Gamma,X)$ the Banach space of all absolutely summable (resp., bounded) families $\{x_{\gamma}\}_{\gamma\in\Gamma}$ of elements of $X$ indexed by $\Gamma$ and equipped with the norm

[TABLE]

(resp.,

[TABLE]

Let $\vec{A}=(A_{0},A_{1})$ be a fixed Banach couple and let $A$ be a fixed intermediate space with respect to $\vec{A}$. If $\vec{X}=(X_{0},X_{1})$ is any Banach couple and $\Gamma:=B_{L(\vec{A},\vec{X})}$ is the unit ball of the Banach space $L(\vec{A},\vec{X})$, then we define a mapping $\pi_{\vec{A}}\colon\ell_{1}(\Gamma,A_{0}+A_{1})\to X_{0}+X_{1}$ by the formula,

[TABLE]

If the unit ball $B_{L(\vec{X},\vec{A})}$ of the Banach space $L(\vec{X},\vec{A})$ is denoted by $J$, for short, we also define an operator $i_{\vec{A}}\colon X_{0}+X_{1}\to\ell_{\infty}(J,A_{0}+A_{1})$ by the formula

[TABLE]

The Aronszajn–Gagliardo exact interpolation functors are defined by

[TABLE]

and

[TABLE]

Norms in these Banach spaces are given by

[TABLE]

and, respectively

[TABLE]

For simplicity, we often write $G^{\vec{A}}_{A}$ (resp., $H^{\vec{A}}_{A}$) instead of $Orb^{\vec{A}}_{A}$ (resp., $Corb^{\vec{A}}_{A}$). Note that $G^{\vec{A}}_{A}$ is the minimal interpolation functor satisfying $A\hookrightarrow G^{\vec{A}}_{A}(\vec{A})$ and $H^{\vec{A}}_{A}$ is the maximal interpolation functor satisfying $H^{\vec{A}}_{A}(\vec{A})\hookrightarrow A$.

We will use the following result.

Theorem 4.1**.**

Assume that $(A,B;C)\in\mathcal{B}_{\varphi}(\vec{A},\vec{B};\vec{C})$ with $\varphi\in\Phi$. Then, for any Banach couples $\vec{X}=(X_{0},X_{1})$, $\vec{Y}=(Y_{0},Y_{1})$ and $\vec{Z}=(Z_{0},Z_{1})$,

[TABLE]

Proof.

Fix $T\colon\vec{X}\times\vec{Y}\to\vec{Z}$. Assume that $(x,y)\in G_{A}^{\vec{A}}(\vec{X})\times G_{B}^{\vec{B}}(\vec{Y})$, where $x=Ua$ and $y=Vb$ with $(a,b)\in A\times B$, $U\colon\vec{A}\to\vec{X}$ and $V\colon\vec{B}\to\vec{Y}$. Then, for a given operator $R\colon\vec{Z}\to\vec{C}$ with $\|R\|_{\vec{Z}\to\vec{C}}\leq 1$, we define a bilinear operator $S_{R}\colon(A_{0}+A_{1})\times(B_{0}+B_{1})\to C_{0}+C_{1}$ by $S_{R}:=RT(U,V)$. Observe that for each $j\in\{0,1\}$ and every $(a_{j},b_{j})\in A_{j}\times B_{j}$,

[TABLE]

This implies that $S_{R}\colon\vec{A}\times\vec{B}\to\vec{C}$ with

[TABLE]

Thus, it follows, by our hypothesis $(A,B;C)\in\mathcal{B}_{\varphi}(\vec{A},\vec{B};\vec{C})$, that

[TABLE]

and so,

[TABLE]

This proves that $T(x,y)=T(Ua,Vb)\in H_{C}^{\vec{C}}(\vec{Z})$ with

[TABLE]

We now assume that $(x,y)\in G_{A}^{\vec{A}}(\vec{X})\times G_{B}^{\vec{B}}(\vec{Y})$ and consider arbitrary representations

[TABLE]

with

[TABLE]

It is clear that the above series converge into $A_{0}+A_{1}$ and $B_{0}+B_{1}$, respectively. It follows (since $T$ is a bilinear operator from $(A_{0}+A_{1})\times(B_{0}+B_{1})$ to $C_{0}+C_{1}$) that the following double series converges into $C_{0}+C_{1}$ to $T(x,y)$,

[TABLE]

Applying estimate $(*)$, we obtain

[TABLE]

Since the representations of $x\in G_{A}^{\vec{A}}(\vec{X})$ and $y\in G_{B}^{\vec{B}}(\vec{Y})$ are arbitrary, we conclude that $T\colon G_{A}^{\vec{A}}(\vec{X})\times G_{B}^{\vec{B}}(\vec{Y})\to H_{C}^{\vec{C}}(\vec{Z})$ with

[TABLE]

This proves that $(G_{A}^{\vec{A}}(\vec{X}),G_{B}^{\vec{B}}(\vec{Y});H_{C}^{\vec{C}}(\vec{Z}))\in\mathcal{B}_{\varphi}(\vec{X},\vec{Y};\vec{Z})$, as required. ∎

To prove the main result of this section, we use the continuous inclusions from [11, Lemmas 2.1 and 3.1]. We state these inclusions for the sake of completeness and convenience of the readers.

Lemma 4.2**.**

Let $I$ be any non-empty set and let $A$ be a Banach space intermediate with respect to a Banach couple $\vec{A}=(A_{0},A_{1})$. Then the following continuous inclusions hold:**

[TABLE]

[TABLE]

with norm less than or equal to $1$.

Following [22], the function $\varphi_{\mathcal{F}}$, which corresponds to an exact interpolation functor $\mathcal{F}$ by the equality

[TABLE]

is called the characteristic function of the functor $\mathcal{F}$. Here $\alpha\mathbb{R}$ denotes $\mathbb{R}$ equipped with the norm $\|\cdot\|_{\alpha\mathbb{R}}=\alpha|\cdot|$ for $\alpha>0$. We notice that $\varphi\in\Phi$.

We omit the simple proof of the following technical fact.

Proposition 4.3**.**

Let $A$ be an intermediate Banach space with respect to a couple $\vec{A}$ of Banach spaces. Then the characteristic function $\varphi_{G}$ of an exact interpolation functor $G:=G_{A}^{\vec{A}}$ is given by

[TABLE]

We will need also the following lemma.

Lemma 4.4**.**

If $\varphi_{\mathcal{F}}$ is a characteristic function of an exact interpolation functor $\mathcal{F}$, then

[TABLE]

where $\vec{\mathcal{B}}$ denotes the class of all Banach couples.

Proof.

Let us fix a Banach couple $\vec{A}$. Then from the minimality property of an orbit functor $G:=G_{A}^{\vec{A}}$ with $A:=\mathcal{F}(\vec{A})$, it follows that, for any Banach couple $\vec{X}$,

[TABLE]

with the norm of the continuous inclusion map less than or equal to $1$. In particular this implies that $\varphi_{G}\geq\varphi_{\mathcal{F}}$, and so,

[TABLE]

Since the characteristic function $\varphi_{G}$ of the functor $G$ satisfies $\varphi_{G}(s,t)=(\psi_{A})^{*}(s,t)$ for all $s,t>0$, we conclude that

[TABLE]

A direct computation shows that, for fixed $u,v>0$ and all $\alpha\in\mathbb{R}$,

[TABLE]

Hence, for all $s,t>0$, we get

[TABLE]

This completes the proof. ∎

Corollary 4.5**.**

Let $\vec{A}=(A_{0},A_{1})$ be a fixed Banach couple and let $A$ be a fixed intermediate space with respect to $\vec{A}$. Then, for any Banach couple $\vec{X}$, the fundamental function $\psi_{X}$ of $X:=Orb_{A}^{\vec{A}}(\vec{X})$ satisfies the estimate

[TABLE]

In particular $\psi_{A}\in\Phi_{0}$ implies $\psi_{X}\in\Phi_{0}$.

Proof.

As mentioned in the proof of Lemma 4.4, the fundamental function of the functor $\mathcal{F}:=Orb_{A}^{\vec{A}}$ satisfies $(\varphi_{F})^{*}=\psi_{A}.$ Applying Lemma 4.4 to the functor $\mathcal{F}$, the required estimate follows (by $X=\mathcal{F}(\vec{X})$)

[TABLE]

∎

We now state and prove the following key theorem, which will be used repeatedly.

Theorem 4.6**.**

Suppose that Banach spaces $A$, $B$, $C$ and Banach couples $\vec{A}=(A_{0},A_{1})$, $\vec{B}=(B_{0},B_{1})$, $\vec{C}=(C_{0},C_{1})$ satisfy the conditions of Theorem 3.2. Let $(X_{0},X_{1})$, $(Y_{0},Y_{1})$, $(Z_{0},Z_{1})$ be any Banach couples and let $T\colon(X_{0},X_{1})\times(Y_{0},Y_{1})\to(Z_{0},Z_{1})$ be a bilinear operator, such that the restriction $T\colon X_{0}\times Y_{0}\to Z_{0}$ is compact. Then

[TABLE]

is also a compact operator.

Proof.

For simplicity, we denote by $I_{0}$, $I_{1}$ and $I$ the balls of the Banach spaces $L(\vec{A},\vec{X})$, $L(\vec{B},\vec{Y})$ and $L(\vec{C},\vec{Z})$, respectively. From Proposition 4.1 combined with the definitions of minimal and maximal interpolation functors, it follows that the operator

[TABLE]

is compact, if and only if, the operator $\widetilde{T}$ given by the formula

[TABLE]

is compact. We thus have the following factorizations for the restrictions

[TABLE]

[TABLE]

Applying Lemma 4.1, we conclude that

[TABLE]

Since the couples $(A_{0},A_{1})$, $(B_{0},B_{1})$ and $(C_{0},C_{1})$ have the approximation property $(Ap)$, then the couples $(\ell_{1}(I_{0},A_{0}),\ell_{1}(I_{0},A_{1}))$, $(\ell_{1}(I_{1},B_{0}),\ell_{1}(I_{1},B_{1}))$ and $(\ell_{1}(I,C_{0}),\ell_{1}(I,C_{1}))$ inherit the same property. Further, our hypothesis $A_{0}=A_{0}^{\circ}$ and $B_{0}=B_{0}^{\circ}$ implies that, for couples $(\ell_{1}(I_{0},A_{0}),\ell_{1}(I_{0},A_{1}))$, $(\ell_{1}(I_{1},B_{0}),\ell_{1}(I_{1},B_{1}))$, we have

[TABLE]

Combining the above facts, we conclude from Theorem 3.2 that

[TABLE]

is a compact operator. Consequently, it follows, from Lemma 4.2,

[TABLE]

is compact. Combining this fact with obvious continuous inclusions

[TABLE]

we conclude that

[TABLE]

is a compact operator as required. The proof is complete. ∎

We conclude this section by specializing Theorem 4.6 to specific couples $\vec{A}$, $\vec{B}$ and $\vec{C}$ satisfying the approximation property $(Ap)$ and triples $(A,B;C)\in\mathcal{B}_{\varphi}(\vec{A},\vec{B}\;\vec{C})$ to get new results on interpolation of bilinear compact operators. We start with applications which involve the Calderón complex method of interpolation $[\,\cdot\,]_{\theta}$ with $\theta\in(0,1)$. Information on this method is found in [8, 5].

Theorem 4.7**.**

Suppose that Banach couples $\vec{A}$, $\vec{B}$ and $\vec{C}$ satisfy the approximation property $(Ap)$. Then for any Banach couples $(X_{0},X_{1})$, $(Y_{0},Y_{1})$, $(Z_{0},Z_{1})$ and any bilinear operator $T\colon(X_{0},X_{1})\times(Y_{0},Y_{1})\to(Z_{0},Z_{1})$ such that $T\colon X_{0}\times Y_{0}\to Z_{0}$ is compact, we have

[TABLE]

is a compact bilinear operator for every $\theta\in(0,1)$.

Proof.

We apply Theorem 4.6. Observe that for any Banach couple $\vec{A}=(A_{0},A_{1})$, we have $[A_{0},A_{1}]_{\theta}\hookrightarrow(A_{0},A_{1})_{\theta,\infty}$ for all $\theta\in(0,1)$ with norm less than or equal to $1$ (see [5, Theorem 4.7.1]). This implies that the fundamental function $\psi_{[\vec{A}]_{\theta}}$ of the space $[\vec{A}]_{\theta}$ satisfies the estimate

[TABLE]

and so $\psi_{[\vec{A}]_{\theta}}\in\Phi_{0}$ for all $\theta\in(0,1)$.

According to multilinear theorem by Calderón (see [8] or [5, Theorem 4.4.1]), it follows that, for any bilinear operator $S\colon\vec{A}\times\vec{B}\to\vec{C}$, we have $S\colon[\vec{A}]_{\theta}\times[\vec{B}]_{\theta}\to[\vec{C}]_{\theta}$ with

[TABLE]

This implies that

[TABLE]

where $\varphi(s,t)=s^{1-\theta}t^{\theta}$ for all $s,t>0.$

Combining the above facts with the well known isometrical formula true for any Banach couple $(X_{0},X_{1})$,

[TABLE]

we see that the required result follows from Theorem 4.6. ∎

Before proceeding applications for bilinear operators on the product of interpolation spaces generated by Peetre’s method $\langle\,\cdot\,\rangle_{\theta}$, we recall that, for any Banach couple $(X_{0},X_{1})$ and every $\theta\in(0,1)$, the space $\langle X_{0},X_{1}\rangle_{\theta}$ is defined as the set of all elements $x\in X_{0}+X_{1}$ which are represented in the form $x=\sum_{k\in\mathbb{Z}}x_{k}$ (convergence in $X_{0}+X_{1}$), where the elements $x_{k}\in X_{0}\cap X_{1}$ are such that $\sum_{k\in\mathbb{Z}}2^{-\theta k}x_{k}$ is unconditionally convergent in $X_{0}$, and $\sum_{k\in\mathbb{Z}}2^{(1-\theta)k}x_{k}$ is unconditionally convergent in $X_{1}$. $\langle X_{0},X_{1}\rangle_{\theta}$ is a Banach space equipped with the norm

[TABLE]

where the supremum takes over all sequences $(\varepsilon_{k})=(\pm 1)$ and the infimum takes over all representations as above $x=\sum_{k\in\mathbb{Z}}x_{k}$.

Couples $(c_{0},c_{0}(2^{-n}))$ and $(\ell_{1},\ell_{1}(2^{-n}))$ of $c_{0}$-spaces and $\ell_{1}$-spaces modelled on $\mathbb{Z}$ are denoted by $\vec{c}_{0}$ and $\vec{\ell}_{1}$. If $\varphi\in\Phi$, then $\ell_{1}(\varphi^{*}(1,2^{-n})$ is an intermediate space between $\ell_{1}$ and $\ell_{1}(2^{-n})$. We denote by $H_{\varphi}$ the Ovchinnikov functor

[TABLE]

If $\varphi(s,t)=s^{1-\theta}t^{\theta}$, for all $s,t>0$ and some $\theta\in(0,1)$, we write $H_{\theta}$ instead of $H_{\varphi}$.

Theorem 4.8**.**

Let $(X_{0},X_{1})$, $(Y_{0},Y_{1})$ and $(Z_{0},Z_{1})$ be Banach couples. Then, for any bilinear operator $T\colon(X_{0},X_{1})\times(Y_{0},Y_{1})\to(Z_{0},Z_{1})$, such that $T\colon X_{0}\times Y_{0}\to Z_{0}$ is compact, we have

[TABLE]

is a compact bilinear operator for every $\theta\in(0,1)$.

Proof.

It is obvious that Banach couples $\vec{c}_{0}$ and $\vec{\ell}_{1}$ satisfy approximation property $(Ap)$. The following well known isometrical formulas

[TABLE]

combined with orbital description of Peetre’s functor (see [15] or [22, p. 468])

[TABLE]

completes the proof by Theorem 4.7 applied for couples $\vec{A}=\vec{B}:=\vec{c}_{0}$ and $\vec{C}:=\vec{\ell}_{1}$. ∎

We will show applications of the above result to Calderón products of Banach function lattices. When the complex method is applied to a couple $(X_{0},X_{1})$ of Banach function lattices, we surmise that $X_{j}:=X_{j}(\mathcal{C})$ is a complexification of $X_{j}$ for each $j=0,1$ on a $\sigma$-finite complete measure space $(\Omega,\Sigma,\mu)$ with $\text{supp}(X_{j})=\Omega$. We recall that the Calderón product space $X_{0}^{1-\theta}X_{1}^{\theta}$ is defined for any couple $(X_{0},X_{1})$ of Banach function lattices on measure space $(\Omega,\Sigma,\mu)$. It consists of all $f\in L^{0}(\mu)$, such that $|f|\leq\lambda\,|f_{0}|^{1-\theta}|f_{1}|^{\theta}$ $\mu$-a.e. for some $\lambda>0$ and $f_{j}\in X_{j}$ with $\|f_{j}\|_{X_{j}}\leq 1$, $j=0,1$. It is well known (see [8]) that $X_{0}^{1-\theta}X_{1}^{\theta}$ is a Banach function lattice equipped with the norm

[TABLE]

As usual for a given Banach function lattice over $(\Omega,\Sigma,\mu)$, by $X^{\prime}$, we denote the Köthe dual space of $X$ of all $f\in L^{0}(\mu)$ equipped with the norm

[TABLE]

A Banach function lattice $X$ has the Fatou property, provided that the unit ball is closed n $L^{0}(\mu)$ equipped with the topology of convergence in measure on $\mu$-finite sets. It is well known that the Fatou property is equivalent to $X^{\prime\prime}=X$, isometrically.

Let us draw a useful conclusion in the setting of Calderón product spaces.

Corollary 4.9**.**

Let $(X_{0},X_{1})$, $(Y_{0},Y_{1})$ and $(Z_{0},Z_{1})$ be Banach function lattices on the corresponding measure spaces. Assume that $T\colon(X_{0},X_{1})\times(Y_{0},Y_{1})\to(Z_{0},Z_{1})$ is a bilinear operator, such that $\colon X_{0}\times Y_{0}\to Z_{0}$ is compact. Then,

[TABLE]

is a compact bilinear operator. In particular,

[TABLE]

is compact whenever $Z_{0}$ and $Z_{1}$ have the Fatou property.

Proof.

For any couple $(E_{0},E_{1})$ of Banach lattices and $\theta\in(0,1)$, we have (see [21, Theorem 2.1])

[TABLE]

and (see [22, Lemma 8.5.1])

[TABLE]

By applying Theorem 4.8, the required statement is given. ∎

We conclude with applications to the real methods of interpolation. Let $E$ be a Banach sequence lattice intermediate with respect to $(\ell_{\infty},\ell_{\infty}(2^{-n}))$. For a given Banach couple $\vec{X}$, we denote by $K_{E}(\vec{X})$ the $K$-space which is the Banach space of all $x\in X_{0}+X_{1}$ such that $\{K(2^{k},x;\vec{X})\}_{k\in\mathbb{Z}}\in E$ equipped with the norm

[TABLE]

It is well known that $K_{E}$ is an exact interpolation functor which is often called $K$-method of interpolation.

We also recall the so called $J$-method of interpolation. As usual for any Banach couple $\vec{X}=(X_{0},X_{1})$, we let $J(t,x;\vec{X}):=\max\{\|x\|_{X_{0}},t\|x\|_{X_{1}}\}$ for any $x\in X_{0}\cap X_{1}$ and all $t>0$. Let $F$ be a Banach sequence lattice intermediate with respect to $(\ell_{1},\ell_{1}(2^{-n}))$. By $J_{F}(\vec{X})$ we denote the $J$-space which is the Banach space of all $x\in X_{0}+X_{1}$ represented in the form

[TABLE]

where $\{J(2^{k},u_{k};\vec{X})\}\in F$ with the norm

[TABLE]

It is well known that $J_{F}$ is an exact interpolation functor.

Observe that $\{J(2^{k},u_{k};\vec{X})\}\in F$ combined with $F\hookrightarrow\ell_{1}+\ell_{1}(2^{-n})$ yields that the series $\sum_{k=-\infty}^{\infty}u_{k}$ converges absolutely into $X_{0}+X_{1}$:

[TABLE]

We note that if a Banach sequence lattice $E$ on $\mathbb{Z}$ satisfies the condition $\ell_{\infty}\cap\ell_{\infty}(2^{-n})\hookrightarrow E\hookrightarrow\ell_{1}+\ell_{1}(2^{-n})$, then $K_{E}(\vec{X})$ and $J_{E}(\vec{X})$ are well defined for any Banach couple $\vec{X}$. This follows immediately from the classical fundamental lemma (see [5])

[TABLE]

Space $E$ is said to be a parameter of the real method if $K_{E}(\vec{X})=J_{E}(\vec{X})$ for any Banach couple $\vec{X}$. It is well known that this is equivalent to the fact that, for any operator $T\colon\vec{\ell}_{1}\to\vec{\ell}_{\infty}$, one has $T\colon E\to E$ (see, e.g., [22, Lemma 7.3.1]).

We are now able to state our general bilinear interpolation theorem on compactness for bilinear operators on real methods spaces.

Theorem 4.10**.**

Let $E_{0}$, $E_{1}$ and $F$ be Banach sequence lattices, such that $(E_{0},E_{1};F)\in\mathcal{B}_{\varphi}(\vec{\ell}_{1},\vec{\ell}_{1};\vec{\ell}_{\infty})$ for some $\varphi\in\Phi_{0}$, and let $\psi_{E_{0}}(s,1)\to 0$, $\psi_{E_{1}}(s,1)\to 0$ and $\phi_{F}(s,1)\to 0$ as $s\to 0$. Then, for any Banach couples $(X_{0},X_{1})$, $(Y_{0},Y_{1})$ and $(Z_{0},Z_{1})$ and any bilinear operator $T\colon(X_{0},X_{1})\times(Y_{0},Y_{1})\to(Z_{0},Z_{1})$ such that $T\colon X_{0}\times Y_{0}\to Z_{0}$ is compact, we obtain

[TABLE]

is a compact bilinear operator.

Proof.

From the well known isometrical description of coorbital (resp., orbital) of the $K$-space (resp., $J$-space), we have, for any Banach couple $(A_{0},A_{1})$ (see [7, Theorems 3.3.4, 3.4.12] or [22, Theorems 7.1.1, 7.2.1]):

[TABLE]

Since $\vec{\ell}_{1}$ is a regular couple,

[TABLE]

Now we are in a position to apply Theorem 4.6 to get the statement. ∎

We provide a result which gives a complete description of triples of Banach sequence lattices $(E_{0},E_{1};F)\in\mathcal{B}_{\varphi}(\vec{\ell}_{1},\vec{\ell}_{1};\vec{\ell}_{\infty})$ in terms of boundedness of the convolution operator $\sigma$ defined on $(\ell_{1}+\ell_{1}(2^{-n}))\times(\ell_{1}+\ell_{1}(2^{-n}))$ by $\sigma(x,y)=x\star y$, for all $x=\{x_{n}\}$ and $y=\{y_{n}\}$ in $\ell_{1}+\ell_{1}(2^{-n})$, where

[TABLE]

If Banach sequence lattices $E_{0}$, $E_{1}$ and $E_{2}$ intermediate with respect to $\vec{\ell}_{1}$ are such that the convolution operator $\sigma\colon E_{0}\times E_{1}\to E_{2}$, then we write $E_{0}\star E_{1}\subset E_{2}$ for short.

At first we prove the following lemma.

Theorem 4.11**.**

Let Banach sequence lattices $E_{0}$, $E_{1}$ and $E_{2}$ be intermediate with respect to $\vec{\ell}_{1}$ such that $E_{0}\star E_{1}\subset E_{2}$. Then for $F=J_{E_{2}}(\vec{\ell}_{\infty})$, we have $(E_{0},E_{1};F)\in\mathcal{B}(\vec{\ell}_{1},\vec{\ell}_{1};\vec{\ell}_{\infty})$. In particular $(E_{0},E_{1};F)\in\mathcal{B}_{\varphi}(\vec{\ell}_{1},\vec{\ell}_{1};\vec{\ell}_{\infty})$ with $\varphi$ defined by

[TABLE]

where the supremum takes over all bilinear operators $T\colon\vec{\ell}_{1}\times\vec{\ell}_{1}\to\vec{\ell}_{\infty}$, such that $\|T\|_{\ell_{1}\times\ell_{1}\to\ell_{\infty}}\linebreak\leq s$ and $\|T\|_{\ell_{1}(2^{-n})\times\ell_{1}(2^{-n})\to\ell_{\infty}(2^{-n})}\leq t$.

Proof.

Let us assume that $E_{0}\star E_{1}\subset E_{2}$ and let $T\colon\vec{\ell}_{1}\times\vec{\ell}_{1}\to\vec{\ell}_{\infty}$ be any bilinear operator with norm less than or equal to $1$.

Fix $x=\{x_{m}\}\in E_{0}$ and $y\in\{x_{k}\}\in E_{1}$. By $E_{j}\hookrightarrow\ell_{1}+\ell_{1}(2^{-n})$ for $j\in\{0,1\}$, then the two series

[TABLE]

converge absolutely in $\ell_{1}+\ell_{1}(2^{-n}))$, where $e_{n}$ denotes the standard unit basis vector for each $n\in\mathbb{Z}$.

Since $T\colon(\ell_{1}+\ell_{1}(2^{-n}))\times(\ell_{1}+\ell_{1}(2^{-n}))\to\ell_{\infty}+\ell_{\infty}(2^{-n})$ is continuous,

[TABLE]

where each double series converges absolutely into $\ell_{\infty}+\ell_{\infty}(2^{-n})$. Consequently,

[TABLE]

with convergence in $\ell_{\infty}+\ell_{\infty}(2^{-n})$.

Observe that for each $k\in\mathbb{Z}$, we have (by $\|T\|_{\vec{\ell}_{1}\times\vec{\ell}_{1}\to\vec{\ell}_{\infty}}\leq 1$)

[TABLE]

and similarly for each $k\in\mathbb{Z}$,

[TABLE]

Combining the above estimates, we conclude that

[TABLE]

and

[TABLE]

Since

[TABLE]

and there exists a positive constant $C$ (since $\sigma$ is positive, $\sigma\colon E_{0}\times E_{1}\to E_{2}$ is a bounded bilinear operator)

[TABLE]

we get that $T(x,y)\in J_{E_{2}}(\vec{\ell}_{\infty})$ with

[TABLE]

This completes the proof of the first statement. The second statement is obvious. ∎

Let us conclude by remarking that the convolution operator $\sigma\in\mathcal{B}(\vec{\ell}_{1},\vec{\ell}_{1};\vec{\ell}_{\infty})$, and so an immediate consequence of Theorem 4.11, is the following result. If $E_{0}$, $E_{1}$ and $F$ are Banach sequence lattices intermediate with respect to $\vec{\ell}_{1}$ and if $F$ is a real parameter of the real method, then $(E_{0},E_{1};F)\in\mathcal{B}(\vec{\ell}_{1},\vec{\ell}_{1};\vec{\ell}_{\infty})$, if and only if, $E_{0}\star E_{1}\subset F$. This observation in combination with Theorem 4.10 in particular yields a more general variant of a bilinear compactness interpolation theorem established in [13, Theorem 3.1] for spaces generated by parameters of the real method.