The Brown–Halmos theorem for discrete Wiener–Hopf operators

Abstract

We prove an analogue of the Brown–Halmos theorem for discrete Wiener–Hopf operators acting on separable rearrangement-invariant Banach sequence spaces.

1 Introduction and the main results

The study of Wiener–Hopf integral operators

$$\begin{aligned} (Tf)(x)=\int _0^\infty K(x-y)f(y)\,dy, \quad x>0, \end{aligned}$$

was started by Wiener and Hopf in 1931. Discrete Wiener–Hopf operators arise naturally as discretizations of their integral counterparts. A fairly complete theory of these operators on Lebesgue spaces $L^p$ and $\ell ^p$, respectively, is presented in several monographs (see, e.g., [4,5,6, 10, 11, 19]). Rearrangement-invariant Banach sequence spaces constitute natural generalizations of the scale $\ell ^p$ with $1\le p\le \infty$. As far as we know, there were no studies of discrete Wiener–Hopf operators in this setting. The aim of this paper is to start investigations in this direction and to establish one of the most basic facts about discrete Wiener–Hopf operators on separable rearrangement-invariant Banach sequence spaces—an analogue of the Brown–Halmos theorem.

For a Banach space $\mathcal {X}$, let $\mathcal {B}(\mathcal {X})$ denote the Banach algebra of all bounded linear operators on $\mathcal {X}$ and let $\Vert A\Vert _{\mathcal {B}(\mathcal {X})}$ be the norm of an operator $A\in \mathcal {B}(\mathcal {X})$.

Let $X(\mathbb {Z})$ be a separable rearrangement-invariant Banach sequence space and let $X'(\mathbb {Z})$ be its associate space (see [3, Ch. 1-2] and Sect. 2.2). The class of rearrangement-invariant Banach sequence spaces includes classical Lebesgue sequence spaces $\ell ^p(\mathbb {Z})$ with $1\le p\le \infty$, Orlicz sequence spaces $\ell ^\Phi (\mathbb {Z})$, and Lorentz sequence spaces $\ell ^{p,q}(\mathbb {Z})$ with $1<p<\infty$, $1\le q\le \infty$, among others.

Let $\mathcal {P}'$ be the space of periodic distributions (see, e.g., [2, Ch. 3 and 5] and Sect. 2.3) and let $S_0(\mathbb {Z})$ denote the set of all finitely supported sequences. For $a\in \mathcal {P}'$ and $\varphi \in S_0(\mathbb {Z})$, we define the convolution $a*\varphi$ as the sequence

$$\begin{aligned} (a*\varphi )_j=\sum _{k\in \mathbb {Z}}\widehat{a}_{j-k}\varphi _k, \quad j\in \mathbb {Z}, \end{aligned}$$

where $(\widehat{a}_j)_{j\in \mathbb {Z}}$ is the sequence of Fourier coefficients of the distribution a. By $M_{X(\mathbb {Z})}$ we denote the collection of all distributions $a\in \mathcal {P}'$ for which $a*\varphi \in X(\mathbb {Z})$ whenever $\varphi \in S_0(\mathbb {Z})$ and

$$\begin{aligned} \Vert a\Vert _{M_{X(\mathbb {Z})}}:=\sup \left\{ \frac{\Vert a*\varphi \Vert _{X(\mathbb {Z})}}{\Vert \varphi \Vert _{X(\mathbb {Z})}}\,\ \varphi \in S_0(\mathbb {Z}), \ \varphi \ne 0 \right\} <\infty . \end{aligned}$$

Since $S_0(\mathbb {Z})$ is dense in $X(\mathbb {Z})$ (see Lemma 3 below), for $a\in M_{X(\mathbb {Z})}$, the operator from $S_0(\mathbb {Z})$ to $X(\mathbb {Z})$ defined by $\varphi \mapsto a*\varphi$ extends to a bounded operator

$$\begin{aligned} L(a):X(\mathbb {Z})\rightarrow X(\mathbb {Z}), \quad \varphi \mapsto a*\varphi , \end{aligned}$$

which is referred to as the Laurent operator with symbol a.

Note that, in the case of $X(\mathbb {Z})=\ell ^p(\mathbb {Z})$ with $1\le p<\infty$, our definition of $M_{\ell ^p(\mathbb {Z})}$ is formally more general than of $M^p$ adapted in [6, Section 2.3, p. 46], where symbols of L(a) are functions in $L^1([0,2\pi ])$ and not distributions. The reason for our choice in this definition is explained after the statement of Theorem 1. Nevertheless, repeating the argument of [6, Section 2.5(d)], one can show that $M_{\ell ^p(\mathbb {Z})}$ is continuously embedded into $L^\infty ([0,2\pi ])$. So, $M^p=M_{\ell ^p(\mathbb {Z})}$.

This reasoning can also be extended to reflexive reflection-invariant Banach sequence spaces $X(\mathbb {Z})$, for which it can be shown that $M_{X(\mathbb {Z})}$ is continuously embedded into $L^\infty ([0,2\pi ])$. Similarly to [6, Section 2.5(g)], one can show that $M_{X(\mathbb {Z})}$ is a Banach algebra. We plan to provide detailed proofs of these facts in a forthcoming publication.

It is known that $M_{\ell ^1(\mathbb {Z})}$ coincides with the Wiener algebra and $M_{\ell ^2(\mathbb {Z})}$ is $L^\infty ([0,2\pi ])$. In all other cases, a reasonable description of $M_{X(\mathbb {Z})}$ is unknown. Hence, one should look for sufficient conditions guaranteeing that $a\in M_{X(\mathbb {Z})}$. For instance, with the aid of the Boyd interpolation theorem (see [7] and also [3, Ch. 3, Theorem 5.16]) and Stechkin’s inequality (see [6, Section 2.5(f)]), one can prove that if $X(\mathbb {Z})$ is a separable rearrangement-invariant Banach sequence space with Boyd indices $\alpha _X,\beta _X$ satisfying $0<\alpha _X\le \beta _X<1$, then any $2\pi$-periodic function of bounded variation belongs to $M_{X(\mathbb {Z})}$.

For $n\in \mathbb {Z}$, let $e_n$ denote the element of $X(\mathbb {Z})$ given by $e_n:=(\delta _{jn})_{j\in \mathbb {Z}}$, where $\delta _{jn}$ is the Kronecker delta. For $f=(f_k)_{k\in \mathbb {Z}}\in X(\mathbb {Z})$ and $g=(g_k)_{k\in \mathbb {Z}}\in X'(\mathbb {Z})$, put

$$\begin{aligned} (f,g)=\sum _{k\in \mathbb {Z}}f_k\overline{g_k}. \end{aligned}$$

For every $a\in M_{X(\mathbb {Z})}$, we have

$$\begin{aligned} (L(a)e_j,e_k)=\widehat{a}_{k-j}, \quad j,k\in \mathbb {Z}. \end{aligned}$$

(1.1)

That is, the matrix of L(a) is doubly-infinite and constant along diagonals:

$$\begin{aligned} \left( \begin{array}{ccc|cccc} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots \\ \ \ddots \ {} &{} \ \widehat{a}_0 \ {} &{} \ \widehat{a}_{-1}\ {} &{} \ \widehat{a}_{-2}\ {} &{} \ \widehat{a}_{-3}\ {} &{} \ \widehat{a}_{-4}\ {} &{} \ \ddots \\ \ \ddots &{} \ \widehat{a}_{1}\ {} &{} \ \widehat{a}_{0}\ {} &{} \ \widehat{a}_{-1}\ {} &{} \ \widehat{a}_{-2}\ {} &{} \ \widehat{a}_{-3}\ {} &{} \ddots \\ \hline \ \ddots \ {} &{} \ \widehat{a}_{2}\ {} &{} \ \widehat{a}_{1}\ {} &{} \ \widehat{a}_{0}\ {} &{} \ \widehat{a}_{-1}\ {} &{} \ \widehat{a}_{-2}\ {} &{} \ddots \\ \ \ddots \ {} &{} \ \widehat{a}_{3}\ {} &{} \ \widehat{a}_{2}\ {} &{} \ \widehat{a}_{1}\ {} &{} \ \widehat{a}_{0}\ {} &{} \ \widehat{a}_{-1}\ {} &{} \ddots \\ \ \ddots \ {} &{} \ \widehat{a}_{4}\ {} &{} \ \widehat{a}_{3}\ {} &{} \ \widehat{a}_{2}\ {} &{} \ \widehat{a}_{1}\ {} &{} \ \widehat{a}_{0}\ {} &{} \ddots \\ \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots \end{array}\right) . \end{aligned}$$

Such structured matrices are usually called Laurent matrices.

Our first result extends [8, Theorem 3] and [6, Proposition 2.4]. It shows that each bounded linear operator on a separable rearrangement-invariant Banach sequence space, whose matrix is constant along diagonals, is a Laurent operator.

Theorem 1

Let $X(\mathbb {Z})$ be a separable rearrangement-invariant Banach sequence space. Suppose A is a bounded linear operator on $X(\mathbb {Z})$ and there is a sequence $(a_n)_{n\in \mathbb {Z}}$ of complex numbers such that

$$\begin{aligned} (Ae_j,e_k)=a_{k-j}, \quad j,k\in \mathbb {Z}. \end{aligned}$$

(1.2)

Then there is a periodic distribution $a\in M_{X(\mathbb {Z})}$ such that $A=L(a)$ and $(a_n)_{n\in \mathbb {Z}}$ is the sequence of Fourier coefficients of a.

Note that the proof of [6, Proposition 2.4] for $\ell ^p(\mathbb {Z})$ is based on the observation that either $\ell ^p(\mathbb {Z})$ or its dual $(\ell ^p(\mathbb {Z}))^*$ is embedded into $\ell ^2(\mathbb {Z})$ and for a given sequence $(a_n)_{n\in \mathbb {Z}}$ in $\ell ^2(\mathbb {Z})$ there is a function $a\in L^2([0,2\pi ])$ such that $a_n=\widehat{a}_n$ for $n\in \mathbb {Z}$. One cannot guarantee that an arbitrary rearrangement-invariant Banach sequence space or its dual is embedded into $\ell ^2(\mathbb {Z})$. However, since $X(\mathbb {Z})\hookrightarrow \ell ^\infty (\mathbb {Z})$, we can circumvent this difficulty by allowing elements of $M_{X(\mathbb {Z})}$ to be periodic distributions and employing Theorem 5 below.

Let $\mathbb {Z}_+:=\{0,1,2,\dots \}$ and let P denote the discrete Riesz projection on $X(\mathbb {Z})$ defined for $\varphi =(\varphi _j)_{j\in \mathbb {Z}}\in X(\mathbb {Z})$ by

$$\begin{aligned} (P\varphi )_j:=\left\{ \begin{array}{ll} \varphi _j &{} \text{ if } j\in \mathbb {Z}_+, \\ 0 &{} \text{ otherwise }. \end{array}\right. \end{aligned}$$

Consider the following subspaces of $X(\mathbb {Z})$:

$$\begin{aligned} S_0(\mathbb {Z}_+)&:=PS_0(\mathbb {Z})= \{\varphi =(\varphi _j)_{j\in \mathbb {Z}}\in S_0(\mathbb {Z}):\varphi _j=0 \text{ for } j\in \mathbb {Z}\setminus \mathbb {Z}_+ \}, \\ X(\mathbb {Z}_+)&:=PX(\mathbb {Z})= \{\varphi =(\varphi _j)_{j\in \mathbb {Z}}\in X(\mathbb {Z}):\varphi _j=0 \text{ for } j\in \mathbb {Z}\setminus \mathbb {Z}_+\}. \end{aligned}$$

Since the subspace $S_0(\mathbb {Z}_+)$ is dense in $X(\mathbb {Z}_ +)$ in view of Lemma 3 below, for every $a\in M_{X(\mathbb {Z})}$, the operator from $S_0(\mathbb {Z}_+)$ to $X(\mathbb {Z}_+)$ defined by $\varphi \mapsto P(a*\varphi )$, extends to a bounded operator

$$\begin{aligned} T(a):X(\mathbb {Z}_+)\rightarrow X(\mathbb {Z}_+), \quad \varphi \mapsto P(a*\varphi ), \end{aligned}$$

which is referred to as the discrete Wiener–Hopf (or Toeplitz) operator with symbol a. Clearly, we have

$$\begin{aligned} \Vert T(a)\Vert _{\mathcal {B}(X(\mathbb {Z}_+))} \le \Vert L(a)\Vert _{\mathcal {B}(X(\mathbb {Z}))}. \end{aligned}$$

(1.3)

For every $a\in M_{X(\mathbb {Z})}$, we have

$$\begin{aligned} (T(a)e_j,e_k)=\widehat{a}_{k-j}, \quad j,k\in \mathbb {Z}_+. \end{aligned}$$

(1.4)

So, the matrix of T(a) on a separable rearrangement-invariant Banach sequence space $X(\mathbb {Z}_+)$ is the infinite Toeplitz matrix

$$\begin{aligned} \left( \begin{array}{cccc} \ \widehat{a}_0 \ {} &{} \ \widehat{a}_{-1}\ {} &{}\ \widehat{a}_{-2}\ {} &{}\ \dots \\ \widehat{a}_1 &{} \widehat{a}_{0} &{} \widehat{a}_{-1} &{} \ddots \\ \widehat{a}_2 &{} \widehat{a}_{1} &{} \widehat{a}_{0} &{} \ddots \\ \vdots &{} \ddots &{} \ddots &{} \ddots \end{array}\right) . \end{aligned}$$

(1.5)

Brown and Halmos [8, Theorem 4] proved that each operator on the Hardy space $H^2(\mathbb {T})$ over the unit circle, whose matrix is (1.5), is a Toeplitz operator on $H^2(\mathbb {T})$. This result was extended by Böttcher and Silbermann [6, Theorem 2.7] to the setting of Toeplitz operators on $H^p(\mathbb {T})$, $1<p<\infty$ and discrete Wiener–Hopf operators on $\ell ^p(\mathbb {Z}_+)$, $1\le p<\infty$ (for the latter, see also Duduchava’s paper [9, Section 1.1]). For further generalizations of the Brown–Halmos theorem for Toeplitz operators on abstract Hardy spaces $H[X(\mathbb {T})]$ built upon Banach function spaces $X(\mathbb {T})$, see [12, 13, 17].

The following version of the Brown–Halmos theorem for discrete Wiener–Hopf operators on rearrangement-invariant Banach sequence spaces is the main result of the paper.

Theorem 2

Let $X(\mathbb {Z})$ be a separable rearrangement-invariant Banach sequence space. Suppose A is a bounded linear operator on the subspace $X(\mathbb {Z}_+)$ and there is a sequence $(a_n)_{n\in \mathbb {Z}}$ of complex numbers such that

$$\begin{aligned} (Ae_j,e_k)=a_{k-j}, \quad j,k\in \mathbb {Z}_+. \end{aligned}$$

(1.6)

Then there is a periodic distribution $a\in M_{X(\mathbb {Z})}$ such that $A=T(a)$ and $(a_n)_{n\in \mathbb {Z}}$ is the sequence of Fourier coefficients of a. Moreover,

$$\begin{aligned} \Vert T(a)\Vert _{\mathcal {B}(X(\mathbb {Z}_+))} = \Vert L(a)\Vert _{\mathcal {B}(X(\mathbb {Z}))} = \Vert a\Vert _{M_{X(\mathbb {Z})}}. \end{aligned}$$

(1.7)

For $X(\mathbb {Z})=\ell ^p(\mathbb {Z})$ with $1\le p<\infty$, the idea of the proof of the above result is indicated on [6, p. 51]. For equality (1.7) in this setting, see also [9, Section 1.1]. We will give a self-contained proof of this result below.

The paper is organized as follows. In Sect. 2, we collect properties of a rearrangement-invariant Banach sequence space $X(\mathbb {Z})$ and its associate space $X'(\mathbb {Z})$. Further, we recall the definitions of the spaces of test functions $\mathcal {P}$ and periodic distributions $\mathcal {P}'$, as well as of the Fourier coefficients of $a\in \mathcal {P}'$. We conclude Sect. 2 with the proof of Young’s convolution inequality for translation-invariant Banach sequence spaces. Section 3 contains complete proofs of Theorems 1 and 2.

2 Preliminaries

2.1 Banach sequence spaces

Let $\ell ^0(\mathbb {Z})$ be the linear space of all sequences $f:\mathbb {Z}\rightarrow \mathbb {C}$, and let $\ell _+^0(\mathbb {Z})$ be the cone of nonnegative sequences in $\ell ^0(\mathbb {Z})$. We equip $\mathbb {Z}$ with the counting measure, i.e. the purely atomic measure with atoms having equal measure 1. According to [3, Ch. 1, Defintion 1.1], a Banach function norm $\varrho :\ell _+^0(\mathbb {Z})\rightarrow [0,\infty ]$ is a mapping which satisfies the following axioms for all $f,g,(f^{(n)})_{n\in \mathbb {N}}$ in $\ell _+^0(\mathbb {Z})$, for all finite subsets $E\subset \mathbb {Z}$, and all constants $\alpha \ge 0$:

$$\begin{aligned} \mathrm{(A1)}{} & {} \varrho (f)=0 \Leftrightarrow f=0, \ \varrho (\alpha f)=\alpha \varrho (f), \ \varrho (f+g) \le \varrho (f)+\varrho (g),\\ \mathrm{(A2)}{} & {} 0\le g \le f \ \ \Rightarrow \ \varrho (g) \le \varrho (f) \quad \text{(the } \text{ lattice } \text{ property) },\\ \mathrm{(A3)}{} & {} 0\le f^{(n)} \uparrow f \ \ \Rightarrow \ \varrho (f^{(n)}) \uparrow \varrho (f)\quad \text{(the } \text{ Fatou } \text{ property) },\\ \mathrm{(A4)}{} & {} \varrho (\chi _E) <\infty ,\\ \mathrm{(A5)}{} & {} \sum _{k\in E} f_k \le C_E\varrho (f), \end{aligned}$$

where $\chi _E$ is the characteristic (indicator) function of E, and the constant $C_E \in (0,\infty )$ may depend on $\varrho$ and E, but is independent of f. The set $X(\mathbb {Z})$ of all sequences $f\in \ell ^0(\mathbb {Z})$ for which $\varrho (|f|)<\infty$ is called a Banach sequence space. For each $f\in X(\mathbb {Z})$, the norm of f is defined by $\Vert f\Vert _{X(\mathbb {Z})}:=\varrho (|f|)$. The set $X(\mathbb {Z})$ equipped with the natural linear space operations and this norm becomes a Banach space (see [3, Ch. 1, Theorems 1.4 and 1.6]). If $\varrho$ is a Banach function norm, its associate norm $\varrho '$ is defined on $\ell _+^0(\mathbb {Z})$ by

$$\begin{aligned} \varrho '(g):=\sup \left\{ \sum _{k\in \mathbb {Z}} f_kg_k \, \ f=(f_k)_{k\in \mathbb {Z}}\in \ell _+^0(\mathbb {Z}), \ \varrho (f) \le 1 \right\} , \quad g\in \ell _+^0(\mathbb {Z}). \end{aligned}$$

It is a Banach function norm itself [3, Ch. 1, Theorem 2.2]. The Banach sequence space $X'(\mathbb {Z})$ determined by the Banach function norm $\varrho '$ is called the associate space (Köthe dual) of $X(\mathbb {Z})$. The associate space $X'(\mathbb {Z})$ can be viewed as a subspace of the Banach dual space $X^*(\mathbb {Z})$.

Let us conclude this subsections with two results highlighting the idea that in many occasions arbitrary sequences can be approximated by finitely supported sequences.

Lemma 3

If $X(\mathbb {Z})$ is a separable Banach sequence space, then $S_0(\mathbb {Z})$ is dense in the space $X(\mathbb {Z})$ and $S_0(\mathbb {Z}_+)$ is dense in the subspace $X(\mathbb {Z}_+)$.

Proof

The first statement of the lemma follows from [3, Ch. 1, Proposition 3.10, Theorem 3.11, and Corollary 5.6]. Let us prove the second part. Fix $\varepsilon >0$. If $f=(f_k)_{k\in \mathbb {Z}}\in X(\mathbb {Z}_+)$, then by the first part there exists $h=(h_k)_{k\in \mathbb {Z}}\in S_0(\mathbb {Z})$ such that $\Vert f-h\Vert _{X(\mathbb {Z})}<\varepsilon$. Take $g=Ph\in S_0(\mathbb {Z}_+)$. Since $f=Pf$ and $\Vert P\Vert _{\mathcal {B}(X(\mathbb {Z}))}=1$, we get

$$\begin{aligned} \Vert f-g\Vert _{X(\mathbb {Z}_+)}=\Vert Pf-Ph\Vert _{X(\mathbb {Z}_+)} \le \Vert f-h\Vert _{X(\mathbb {Z})}<\varepsilon , \end{aligned}$$

which completes the proof. $\square$

Lemma 4

(see [16, Lemma 2.1] and also [14, Lemma 2.10]) Let $X(\mathbb {Z})$ be a Banach sequence space and $X'(\mathbb {Z})$ be its associate space. For every $f=(f_k)_{k\in \mathbb {Z}}\in X(\mathbb {Z})$,

$$\begin{aligned} \Vert f\Vert _{X(\mathbb {Z})} = \sup \left\{ \left| \sum _{k\in \mathbb {Z}} f_ks_k\right| \,\ s=(s_k)_{k\in \mathbb {Z}}\in S_0(\mathbb {Z}),\ \Vert s\Vert _{X'(\mathbb {Z})}\le 1\right\} . \end{aligned}$$

2.2 Rearrangement-invariant Banach sequence spaces

The distribution function of a sequence $f=(f_k)_{k\in \mathbb {Z}}\in \ell ^0(\mathbb {Z})$ is defined by

$$\begin{aligned} d_f(\lambda ):={\text {card}}\{k\in \mathbb {Z}:|f_k|>\lambda \}, \quad \lambda \ge 0, \end{aligned}$$

where ${\text {card}}(\mathbb {S})$ denotes the cardinality of the set $\mathbb {S}\subset \mathbb {Z}$. One says that sequences $f=(f_k)_{k\in \mathbb {Z}},g=(g_k)_{k\in \mathbb {Z}}\in \ell ^0(\mathbb {Z})$ are equimeasurable if $d_f=d_g$. A Banach function norm $\varrho :\ell _+^0(\mathbb {Z})\rightarrow [0,\infty ]$ is said to be rearrangement-invariant if $\varrho (f)=\varrho (g)$ for every pair of equimeasurable sequences $f=(f_k)_{k\in \mathbb {Z}},g=(g_k)_{k\in \mathbb {Z}}\in \ell _+^0(\mathbb {Z})$. In that case, the Banach sequence space $X(\mathbb {Z})$ generated by $\varrho$ is said to be a rearrangement-invariant Banach sequence space (cf. [3, Ch. 2, Definition 4.1]). It follows from [3, Ch. 2, Proposition 4.2] that if a Banach sequence space $X(\mathbb {Z})$ is rearrangement-invariant, then its associate space $X'(\mathbb {Z})$ is also a rearrangement-invariant Banach sequence space.

If $X(\mathbb {Z})$ is a rearrangement-invariant Banach sequence space, then

$$\begin{aligned} \ell ^1(\mathbb {Z}) \hookrightarrow X(\mathbb {Z}) \hookrightarrow \ell ^\infty (\mathbb {Z}), \end{aligned}$$

(2.1)

where $\hookrightarrow$ denotes the continuous embedding (see [3, Ch. 2, Corollary 6.8]). For each $t\in \mathbb {Z}_+$, let $E\subset \mathbb {Z}$ be a set with ${\text {card}}E=t$ and let

$$\begin{aligned} \Phi _X(t)=\Vert \chi _E\Vert _{X(\mathbb {Z})}. \end{aligned}$$

The function so defined is called the fundamental function of $X(\mathbb {Z})$ (see [3, Ch. 2, Definition 5.1]). It is clear that for every $n\in \mathbb {Z}$, one has

$$\begin{aligned} \Vert e_n\Vert _{X(\mathbb {Z})}=\Phi _X(1)<\infty . \end{aligned}$$

2.3 Periodic distributions and their Fourier coefficients

Let $\mathcal {P}$ be the set of all infinitely differentiable $2\pi$-periodic functions from $\mathbb {R}$ to $\mathbb {C}$. Elements of $\mathcal {P}$ are called periodic test functions. One can equip $\mathcal {P}$ with the countable family of seminorms

$$\begin{aligned} \Vert u\Vert _{k,\mathcal {P}}:=\sup _{x\in [0,2\pi ]}|D^{k-1}u(x)|, \quad k\in \mathbb {N}, \end{aligned}$$

where $D^ku$ denotes the k-th derivative of u and $D^0u=u$, and the metric

$$\begin{aligned} d(u,v):=\sum _{k=1}^\infty \frac{1}{2^k} \frac{\Vert u-v\Vert _{k,\mathcal {P}}}{1+\Vert u-v\Vert _{k,\mathcal {P}}}. \end{aligned}$$

Then the set $\mathcal {P}$ endowed with the metric d is a complete linear metric space (see [2, Ch. 3, Theorems 2.1-$-$2.2]).

A periodic distribution is a continuous linear functional on the complete linear metric space $(\mathcal {P},d)$. The set of all periodic distributions is denoted by $\mathcal {P}'$. The functions

$$\begin{aligned} E_n(x):=e^{i n x}, \quad x\in \mathbb {R}, \end{aligned}$$

belong to $\mathcal {P}$ for all $n\in \mathbb {Z}$. The Fourier coefficients of a periodic distribution $a\in \mathcal {P}'$ are defined by

$$\begin{aligned} \widehat{a}_n:=a(E_{-n}), \quad n\in \mathbb {Z}. \end{aligned}$$

A sequence $(a_n)_{n\in \mathbb {Z}}$ is said to be of slow growth if there are some positive constants c and r such that $|a_n|\le c|n|^r$ for all $n\in \mathbb {Z}\setminus \{0\}$. The set of all sequences of slow growth is denoted by $S'(\mathbb {Z})$. It is clear that $\ell ^\infty (\mathbb {Z})\subset S'(\mathbb {Z})$.

The following theorem plays a crucial role in the proof of our main results.

Theorem 5

(see [2, Ch. 5, Theorem 1.2]) A sequence $(a_n)_{n\in \mathbb {Z}}$ of complex numbers is the sequence of Fourier coefficients of a periodic distribution if and only if it is of slow growth.

2.4 Young’s convolution inequality

Let $X(\mathbb {Z})$ be a Banach sequence space and let T be the translation operator defined for $\varphi =(\varphi _j)_{j\in \mathbb {Z}}\in X(\mathbb {Z})$ by

$$\begin{aligned} (T\varphi )_j=\varphi _{j+1}, \quad j\in \mathbb {Z}. \end{aligned}$$

The space $X(\mathbb {Z})$ is said to be translation-invariant if $\Vert T\varphi \Vert _{X(\mathbb {Z})}=\Vert \varphi \Vert _{X(\mathbb {Z})}$ for all $\varphi \in X(\mathbb {Z})$. In this case, the operator T is bounded and invertible on $X(\mathbb {Z})$ and

$$\begin{aligned} (T^{-1}\varphi )_j=\varphi _{j-1}, \quad j\in \mathbb {Z}. \end{aligned}$$

Let $T^0:=I$. For $n\in \mathbb {N}$, let $T^n:=T T^{n-1}$ and $T^{-n}:=(T^{-1})^n$.

It is easy to see that for every sequence $f=(f_j)_{j\in \mathbb {Z}}\in \ell ^0(\mathbb {Z})$, the sequences f and Tf are equimeasurable. Therefore each rearrangement-invariant Banach sequence space is also translation-invariant.

The following statement is a version of Young’s convolution inequality for translation-invariant Banach sequence spaces. Although it should be known, we were not able to find a precise reference (cf. [15, Lemma 3.2] and [18, Lemma 3.1]). We give a proof here for the sake of completeness of presentation.

Theorem 6

Let $X(\mathbb {Z})$ be a translation-invariant Banach sequence space. If $a=(a_j)_{j\in \mathbb {Z}}\in \ell ^1(\mathbb {Z})$ and $b=(b_j)_{j\in \mathbb {Z}}\in X(\mathbb {Z})$, then $a*b\in X(\mathbb {Z})$ and

$$\begin{aligned} \Vert a*b\Vert _{X(\mathbb {Z})} \le \Vert a\Vert _{\ell ^1(\mathbb {Z})}\Vert b\Vert _{X(\mathbb {Z})}, \end{aligned}$$

where

$$\begin{aligned} (a*b)_j=\sum _{k\in \mathbb {Z}}a_{j-k}b_k, \quad j\in \mathbb {Z}. \end{aligned}$$

Proof

For every $c=(c_k)_{k\in \mathbb {Z}}\in X'(\mathbb {Z})$, in view of Tonelli’s theorem (see, e.g., [1, Theorem 5.28]) and Hölder’s inequality for Banach sequence spaces (see [3, Ch. 1, Theorem 2.4]), one has

$$\begin{aligned} \sum _{j\in \mathbb {Z}} |(a*b)_jc_j|&\le \sum _{j\in \mathbb {Z}}\sum _{k\in \mathbb {Z}}|a_{j-k}|\,|b_k|\,|c_j| = \sum _{j\in \mathbb {Z}}\sum _{k\in \mathbb {Z}}|a_k|\,|b_{j-k}|\,|c_j| \\&= \sum _{k\in \mathbb {Z}}|a_k|\left( \sum _{j\in \mathbb {Z}}|b_{j-k}|\,|c_j|\right) = \sum _{k\in \mathbb {Z}}|a_k|\left( \sum _{j\in \mathbb {Z}}|(T^{-k}b)_j|\,|c_j|\right) \\&\le \sum _{k\in \mathbb {Z}}|a_k| \,\Vert T^{-k}b\Vert _{X(\mathbb {Z})}\Vert c\Vert _{X'(\mathbb {Z})}. \end{aligned}$$

Since $X(\mathbb {Z})$ is translation-invariant, we have $\Vert T^{-k}b\Vert _{X(\mathbb {Z})}=\Vert b\Vert _{X(\mathbb {Z})}$ for all $k\in \mathbb {Z}$. Hence, the above inequality implies that

$$\begin{aligned} \sum _{j\in \mathbb {Z}} |(a*b)_jc_j| \le \Vert a\Vert _{\ell ^1(\mathbb {Z})}\Vert b\Vert _{X(\mathbb {Z})}\Vert c\Vert _{X'(\mathbb {Z})}. \end{aligned}$$

In view of the Lorentz–Luxemburg theorem (see [3, Ch. 1, Theorem 2.7]), the last inequality implies that

$$\begin{aligned} \Vert a*b\Vert _{X(\mathbb {Z})}&= \Vert a*b\Vert _{X''(\mathbb {Z})} \\&= \sup \left\{ \sum _{j\in \mathbb {Z}}|(a*b)_jc_j|\ : \ c=(c_j)_{j\in \mathbb {Z}}\in X'(\mathbb {Z}), \ \Vert c\Vert _{X'(\mathbb {Z})}\le 1 \right\} \\&\le \Vert a\Vert _{\ell ^1(\mathbb {Z})}\Vert b\Vert _{X(\mathbb {Z})}, \end{aligned}$$

which completes the proof. $\square$

3 Proofs of the main results

3.1 Proof of Theorem 1

Let $\beta :=Ae_0\in X(\mathbb {Z})$. Then it follows from (1.2) that

$$\begin{aligned} \beta _n=(Ae_0)_n=(Ae_0,e_n)=a_n, \quad n\in \mathbb {Z}, \end{aligned}$$

and hence $\beta =(a_n)_{n\in \mathbb {Z}}$. Since $X(\mathbb {Z})\hookrightarrow \ell ^\infty (\mathbb {Z})\subset S'(\mathbb {Z})$ (see (2.1)), it follows from Theorem 5 that there exists a periodic distribution $a\in \mathcal {P}'$ whose Fourier coefficients sequence is $(a_n)_{n\in \mathbb {Z}}$.

If $\varphi \in S_0(\mathbb {Z})$, then

$$\begin{aligned} \varphi =\sum _{j=m_1}^{m_2}\gamma _j e_j \end{aligned}$$

for some $m_1,m_2\in \mathbb {Z}$ satisfying $m_1\le m_2$ and some $\gamma _j\in \mathbb {C}$, where $j\in \{m_1,\dots ,m_2\}$. Then it follows from (1.1), (1.2) and the equality

$$\begin{aligned} a_n=\widehat{a}_n, \quad n\in \mathbb {Z}, \end{aligned}$$

(3.1)

that for all $k\in \mathbb {Z}$,

$$\begin{aligned} (L(a)\varphi )_k&= \left( \sum _{j=m_1}^{m_2}\gamma _j L(a)e_j \right) _k = \sum _{j=m_1}^{m_2}\gamma _j (L(a)e_j)_k = \sum _{j=m_1}^{m_2}\gamma _j (L(a)e_j,e_k) \\&= \sum _{j=m_1}^{m_2}\gamma _j \widehat{a}_{k-j} = \sum _{j=m_1}^{m_2}\gamma _j a_{k-j} = \sum _{j=m_1}^{m_2}\gamma _j (Ae_j,e_k) \\&= \sum _{j=m_1}^{m_2}\gamma _j (Ae_j)_k = \left( \sum _{j=m_1}^{m_2}\gamma _j Ae_j \right) _k = (A\varphi )_k. \end{aligned}$$

Hence $L(a)\varphi =A\varphi$ for all $\varphi \in S_0(\mathbb {Z})$.

Since $X(\mathbb {Z})$ is separable, $S_0(\mathbb {Z})$ is dense in $X(\mathbb {Z})$ by Lemma 3. Therefore $A=L(a)$ on $X(\mathbb {Z})$ and

$$\begin{aligned} \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}))}&= \sup \left\{ \frac{ \Vert A\varphi \Vert _{X(\mathbb {Z})} }{ \Vert \varphi \Vert _{X(\mathbb {Z})} }\ :\ \varphi \in S_0(\mathbb {Z}), \ \varphi \ne 0\right\} \\&= \sup \left\{ \frac{ \Vert a*\varphi \Vert _{X(\mathbb {Z})} }{ \Vert \varphi \Vert _{X(\mathbb {Z})} }\ :\ \varphi \in S_0(\mathbb {Z}), \ \varphi \ne 0\right\} = \Vert a\Vert _{M_{X(\mathbb {Z})}}. \end{aligned}$$

By the hypothesis, $A\in \mathcal {B}(X(\mathbb {Z}))$. Thus $a\in M_{X(\mathbb {Z})}$ and (3.1) holds. $\square$

3.2 Proof of Theorem 2

Let us show that the sequence $\beta =(a_n)_{n\in \mathbb {Z}}$ belongs to $X(\mathbb {Z})$. For $m\in \mathbb {Z}$, let $\mathbbm {1}_{[m,\infty )}:\mathbb {Z}\rightarrow \mathbb {C}$ be defined by

$$\begin{aligned} \left( \mathbbm {1}_{[m,\infty )}\right) _j = \left\{ \begin{array}{ccc} 1 &{} \text{ if } &{} j\in [m,\infty ), \\ 0 &{} \text{ if } &{} j\notin [m,\infty ). \end{array}\right. \end{aligned}$$

Fix $n\in \mathbb {Z}_+$. Then it follows from (1.6) that for $k\in \mathbb {Z}_+$,

$$\begin{aligned} \left( T^{-n}(\mathbbm {1}_{[-n,\infty )}\beta )\right) _k&= \left( \mathbbm {1}_{[0,\infty )}T^{-n}\beta \right) _k = \left( \mathbbm {1}_{[0,\infty )}\right) _k\left( T^{-n}\beta \right) _k \\&= \left( \mathbbm {1}_{[0,\infty )}\right) _k a_{k-n} = \left( \mathbbm {1}_{[0,\infty )}\right) _k (Ae_n,e_k) \\&= \left( \mathbbm {1}_{[0,\infty )}\right) _k (Ae_n)_k = \left( \mathbbm {1}_{[0,\infty )}Ae_n\right) _k . \end{aligned}$$

It is obvious that for $k\in \mathbb {Z}\setminus \mathbb {Z}_+$,

$$\begin{aligned} \left( T^{-n}(\mathbbm {1}_{[-n,\infty )}\beta )\right) _k = \left( \mathbbm {1}_{[0,\infty )}T^{-n}\beta \right) _k = \left( \mathbbm {1}_{[0,\infty )}Ae_n\right) _k =0. \end{aligned}$$

Hence

$$\begin{aligned} T^{-n}(\mathbbm {1}_{[-n,\infty )}\beta )=\mathbbm {1}_{[0,\infty )}Ae_n. \end{aligned}$$

Taking into account that $X(\mathbb {Z})$ is translation-invariant, we obtain

$$\begin{aligned} \left\| \mathbbm {1}_{[-n,\infty )}\beta \right\| _{X(\mathbb {Z})}&= \left\| T^{-n}(\mathbbm {1}_{[-n,\infty )}\beta )\right\| _{X(\mathbb {Z})} = \left\| \mathbbm {1}_{[0,\infty )}Ae_n\right\| _{X(\mathbb {Z})} = \Vert Ae_n\Vert _{X(\mathbb {Z}_+)} \\&\le \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))}\Vert e_n\Vert _{X(\mathbb {Z}_+)} = \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))}\Phi _X(1)<\infty . \end{aligned}$$

Since $(\mathbbm {1}_{[-n,\infty )}\beta )_j\rightarrow a_j$ for all $j\in \mathbb {Z}$ as $n\rightarrow \infty$, by the Fatou lemma (see [3, Ch. 1, Theorem 1.7(iii)], we get

$$\begin{aligned} \Vert \beta \Vert _{X(\mathbb {Z})} \le \liminf _{n\rightarrow \infty } \left\| \mathbbm {1}_{[-n,\infty )}\beta \right\| _{X(\mathbb {Z})} \le \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))}\Phi _X(1)<\infty . \end{aligned}$$

Thus $\beta =(a_n)_{n\in \mathbb {Z}}\in X(\mathbb {Z})$.

Now define the operator $B:S_0(\mathbb {Z})\rightarrow \ell ^0(\mathbb {Z})$ by $B\varphi =\beta *\varphi$, that is,

$$\begin{aligned} (B\varphi )_j:=\sum _{k\in \mathbb {Z}}a_{j-k}\varphi _k, \quad j\in \mathbb {Z}. \end{aligned}$$

(3.2)

Since $\varphi =(\varphi _n)_{n\in \mathbb {Z}}\in S_0(\mathbb {Z})\subset \ell ^1(\mathbb {Z})$ and $\beta =(a_n)_{n\in \mathbb {Z}}\in X(\mathbb {Z})$, it follows from Young’s convolution inequality (see Theorem 6) that

$$\begin{aligned} \Vert B\varphi \Vert _{X(\mathbb {Z})}=\Vert \beta *\varphi \Vert _{X(\mathbb {Z})} \le \Vert \beta \Vert _{X(\mathbb {Z})}\Vert \varphi \Vert _{\ell ^1(\mathbb {Z})}. \end{aligned}$$

So, for every $\varphi \in S_0(\mathbb {Z})$, one has $B\varphi \in X(\mathbb {Z})$.

Now we will show that B is bounded from the subspace $S_0(\mathbb {Z})$ of the space $X(\mathbb {Z})$ to the space $X(\mathbb {Z})$.

If $j,k\in \mathbb {Z}$, then

$$\begin{aligned} (Be_j,e_k) = (Be_j)_k = \sum _{i\in \mathbb {Z}}a_{k-i}(e_j)_i = a_{k-j}. \end{aligned}$$

(3.3)

For $n\ge \max \{-j,-k\}$, put

$$\begin{aligned} A_ n:=T^{-n}AT^n \end{aligned}$$

It follows from (1.6) and (3.3) that for all $n\ge \max \{-j,-k\}$,

$$\begin{aligned} (A_ne_j,e_k)&= (T^{-n}AT^ne_j,e_k) = (AT^ne_j,T^ne_k) \nonumber \\&= (Ae_{j+n},e_{k+n}) = a_{k-j} = (Be_j,e_k). \end{aligned}$$

(3.4)

Now let $\varphi ,\psi \in S_0(\mathbb {Z})$. Then there exist numbers $m_1,m_2,m_3,m_4\in \mathbb {Z}$ such that $m_1\le m_2$, $m_3\le m_4$ and numbers $\lambda _j\in \mathbb {C}$ with $j\in \{m_1,\dots ,m_2\}$ and $\mu _k\in \mathbb {C}$ with $k\in \{m_3,\dots ,m_4\}$ such that

$$\begin{aligned} \varphi =\sum _{j=m_1}^{m_2}\lambda _je_j, \quad \psi =\sum _{k=m_3}^{m_4}\mu _k e_k. \end{aligned}$$

If $n\ge \max \{-m_1,-m_3\}$, then $n+j,n+k\ge 0$ for all $j\in \{m_1,\dots ,m_2\}$ and $k\in \{m_3,\dots ,m_4\}$. It follows from (3.4) that

$$\begin{aligned} (A_n\varphi ,\psi )&= \sum _{j=m_1}^{m_2} \sum _{k=m_3}^{m_4}\lambda _j\overline{\mu _k}(A_ne_j,e_k) \\&= \sum _{j=m_1}^{m_2} \sum _{k=m_3}^{m_4}\lambda _j\overline{\mu _k}(Be_j,e_k) = (B\varphi ,\psi ). \end{aligned}$$

Hence, in view of Hölder’s inequality for Banach sequence spaces (see [3, Ch. 1, Theorem 2.4]) and the fact that $X(\mathbb {Z})$ is translation-invariant, we have

$$\begin{aligned} |(B\varphi ,\psi )|&\le \limsup _{n\rightarrow \infty }|(A_n\varphi ,\psi )| \\&\le \limsup _{n\rightarrow \infty } \Vert T^{-n} AT^n\varphi \Vert _{X(\mathbb {Z})}\Vert \psi \Vert _{X'(\mathbb {Z})} \\&= \limsup _{n\rightarrow \infty } \Vert AT^n\varphi \Vert _{X(\mathbb {Z})}\Vert \psi \Vert _{X'(\mathbb {Z})} \\&\le \limsup _{n\rightarrow \infty } \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))} \Vert T^n\varphi \Vert _{X(\mathbb {Z_+})}\Vert \psi \Vert _{X'(\mathbb {Z})} \\&= \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))} \Vert \varphi \Vert _{X(\mathbb {Z})}\Vert \psi \Vert _{X'(\mathbb {Z})}. \end{aligned}$$

It follows from the above inequality and Lemma 4 that for all $\varphi \in S_0(\mathbb {Z})$,

$$\begin{aligned} \Vert B\varphi \Vert _{X(\mathbb {Z})}&= \sup \left\{ |(B\varphi ,\psi )|\ :\ \psi \in S_0(\mathbb {Z}), \ \Vert \psi \Vert _{X'(\mathbb {Z})}\le 1\right\} \\&\le \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))} \Vert \varphi \Vert _{X(\mathbb {Z})}. \end{aligned}$$

Since $X(\mathbb {Z})$ is separable, by Lemma 3, $S_0(\mathbb {Z})$ is dense in $X(\mathbb {Z})$. So the linear mapping $B:S_0(\mathbb {Z})\rightarrow X(\mathbb {Z})$ defined by (3.2) can be extended to an operator $B\in \mathcal {B}(X(\mathbb {Z}))$ so that

$$\begin{aligned} \Vert B\Vert _{\mathcal {B}(X(\mathbb {Z}))} \le \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))}. \end{aligned}$$

(3.5)

Since B satisfies (3.3) for all $j,k\in \mathbb {Z}$, it follows from Theorem 1 that there exists a distribution $a\in M_{X(\mathbb {Z})}$ such that $B=L(a)$ and

$$\begin{aligned} \widehat{a}_n=a_n, \quad n\in \mathbb {Z}. \end{aligned}$$

(3.6)

As in the proof of Theorem 1, we can prove that (1.4), (1.6) and (3.6) yield that

$$\begin{aligned} T(a)\varphi =A\varphi \quad \text{ for } \text{ all }\quad \varphi \in S_0(\mathbb {Z}_+). \end{aligned}$$

By Lemma 3, $S_0(\mathbb {Z}_+)$ is dense in $X(\mathbb {Z}_+)$. Thus $T(a)=A$ on $X(\mathbb {Z}_+)$. Now (3.5) implies that

$$\begin{aligned} \Vert a\Vert _{M_{X(\mathbb {Z})}} = \Vert L(a)\Vert _{\mathcal {B}(X(\mathbb {Z}))} = \Vert B\Vert _{\mathcal {B}(X(\mathbb {Z}))} \le \Vert A\Vert _{\mathcal {B}(X(\mathbb {Z}_+))} = \Vert T(a)\Vert _{\mathcal {B}(X(\mathbb {Z}_+))}. \end{aligned}$$

Combining this inequality with (1.3), we arrive at (1.7). $\square$