Kernel Thesis Reproduction and Truncated Toeplitz Operators

Zakieldeen Aboabuda Mohamed Alhassn Ali¹, Amna Mahmoud Ahmed Bakhit¹, Isra Abdalhleem Hassan Ali¹, Manal Yagoub Ahmed Juma², Tahani Mahmoud Ahmed Mohamed¹, Sumia Izzeldeen Abdallah Abdelmajeed³

¹Deanship of the Preparatory Year, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia.
²Department of Mathematic, Faculty of Science, University of Qassim, Buraidah, Saudi Arabia.
³Department of Mathematic, Gezira Collage of Technology, Khartoum, Sudan.

DOI: 10.4236/jamp.2023.1112257                

Abstract

Important operator characteristics (such boundedness or compactness) for particular classes of operators on particular reproducing kernel Hilbert spaces may be impacted by the behaviour of the operators on the reproducing kernels. These results have been shown for Toeplitz operators on the Paley-Wiener space, a reproducing kernel Hilbert space over C. Furthermore, we show how the norm of such an operator has no relation to the supremum of the norms of the pictures of the normalization reproducing kernels of the space. As a result, if this supremum is finite, the operator is implicitly bounded. To further demonstrate that the operator norm is not the same as the supremum of the norms of the pictures of the real normalized reproducing kernels, another example is also provided. We also set out a necessary and sufficient condition for the operators’ compactness in terms of their limiting function on the reproducing kernels.

Keywords

Toeplitz Operators, Compact Operator, The Reproducing Kernel

1. Introduction and Notation

The “Bonsall’s Theorem” implies that the supremum of the norms of the images of the normalization reproducing kernels of the Hardy space in [1] and [2] corresponds to the norm of a Hankel operator on the Hardy space of the disc. This statement may be used to the “Reproducing Kernel Thesis” (in [3] ), it implies that the operation of the operators on the kernels defines important characteristics of certain classes of operators on replicating kernel Hilbert spaces. Huge Hankel operators on the Hardy space of the bidisc and Hankel operators on the Bergman space of the disc both produced results that were equivalent in [4] and [5] , respectively. The replication of kernels for Hankel operators on an extensive group of function spaces are shown to determine boundedness, but there doesn’t seem to be a single theorem that proves this. Actually, it has been shown in [6] that the comparable claim for tiny Hankel operators on the Hardy space of the bidisc is not true. We will prove an equivalent of Bonsall’s Theorem for Toeplitz operators acting on the function space Paley-Wiener.

For more details on the value of this space in signal processing, in [7] . It will be established later. Let $P_E$ is orthogonal projection onto E, $E^{\perp }$ is orthogonal complement of E for E, a closed subspace of a Hilbert space. Let $L^2\left(J\right)$ be the closed subspace of $L^2\left(ℝ\right)$ containing those functions that vanish, or are outside of J, for J, a measurable subset of $ℝ$ , the real line, and ${\chi }_J$ convey the characteristic function of J. We define in $L^2\left(ℝ\right)$ ) inner product by $\langle ,\rangle$ , as well as the $L^2\left(ℝ\right)$ function and operator norms by $\Vert \Vert$ . Other $L^2\left(ℝ\right)$ norms will be identified by the symbol ${\Vert \Vert }_p$ . The (unitary) Fourier transform and its inverse will be denoted by the following definitions and notations:

$\mathcal{F}f\left(x\right)=\hat{f}\left(x\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{ℝ}f}\left(t\right){\text{e}}^{-ixt}\text{d}t,$

${\mathcal{F}}^{-1}f\left(t\right)=\tilde{f}\left(t\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{ℝ}f}\left(x\right){\text{e}}^{itx}\text{d}x.$

2. Hankel Operators in the Hardy Space

Suppose ${\mathcal{H}}_{-}$ and ${\mathcal{H}}_{+}$ are lower and half upper planes of $ℂ$ . We define $H^2\left({\mathcal{H}}_{+}\right)$ is the Hardy space of the upper half plane. By the theorem of Paley–Wiener, $H^2\left({\mathcal{H}}_{+}\right)={\mathcal{F}}^{-1}\left(L^2\left[0,\infty \right)\right)$ , in [8] . Therefore, for $f\in L^2\left(ℝ\right)$ ,

$P_{H^2\left({\mathcal{H}}_{+}\right)}f={\mathcal{F}}^{-1}\left({\chi }_{\left[0,\infty \right)}\hat{f}\right),P_{H^2{\left({\mathcal{H}}_{+}\right)}^{\perp }}f={\mathcal{F}}^{-1}\left({\chi }_{\left(-\infty ,0\right]}\hat{f}\right).$ (1)

If $W\left(t\right)=\sqrt{\pi }\left(t+i\right)$ . For $\varphi \in W{L}^2\left(ℝ\right)$ , let ${\Gamma }_{\varphi }$ is a Hankel operator on $H^2\left({\mathcal{H}}_{+}\right)$ with symbol $\varphi$ so:

${\Gamma }_{\varphi }:H^2\left({\mathcal{H}}_{+}\right)\to H^2{\left({\mathcal{H}}_{+}\right)}^{\perp },{\Gamma }_{\varphi }f=P_{H^2{\left({\mathcal{H}}_{+}\right)}^{\perp }}\left(\varphi f\right).$

See [9] for more information on these operators. It is fundamental that $\varphi \in W{L}^2\left(ℝ\right)$ for ${\Gamma }_{\varphi }$ to be defined, since $\frac{1}{W}\in H^2\left({\mathcal{H}}_{+}\right)$ . Note that under this condition, ${\Gamma }_{\varphi }$ is at least defined on $\frac{1}{W}{H}^{\infty }\left({\mathcal{H}}_{+}\right)$ , a dense subspace of $H^2\left({\mathcal{H}}_{+}\right)$ , also:

$L^2\left(ℝ\right)\cup L^{\infty }\left(ℝ\right)\subseteq W{L}^2\left(ℝ\right)$ .

The following theorem provides both essential and enough criteria for ${\Gamma }_{\varphi }$ to be bounded and closed (also known as “Theorem of Bonsall’s”).

Theorem (2.1): For $w\in {\mathcal{H}}_{+}$ , let $j_w$ be $H^2\left({\mathcal{H}}_{+}\right)$ normalized Replication of kernel connected to w, that is

$j_w\left(t\right)=\frac{i\sqrt{\mathrm{Im}w}}{\sqrt{\pi }\left(t-\bar{w}\right)}.$

Then, for $\varphi \in W{L}^2\left(ℝ\right)$ , ${\Gamma }_{\varphi }$ is limited if and only if it has limits on every $\left\{j_w:w\in {\mathcal{H}}_{+}\right\}$ . Additionally, there is a universal constant M that exists such that

$\Vert {\Gamma }_{\varphi }\Vert \le M\sup \left\{\Vert {\Gamma }_{\varphi }{j}_w\Vert :w\in {\mathcal{H}}_{+}\right\}.$

In addition, ${\Gamma }_{\varphi }$ is compact if and only if

$\underset{w\in {\mathcal{H}}_{+},\left|w\right|\to \infty }{\lim }\Vert {\Gamma }_{\varphi }{j}_w\Vert =\underset{w\in {\mathcal{H}}_{+},\mathrm{Im}w\to 0}{\lim }\Vert {\Gamma }_{\varphi }{j}_w\Vert =0.$

The boundedness result is proved in [1] for Hankel operators on the Hardy space of the disc by using Fefferman’s duality theorem. The boundedness result for Hankel operators on the Hardy space of the disc is shown in [1] by using Fefferman’s duality theorem. The compactness result is a simple corollary, which is quoted in [4] , for instance. This version may then be obtained by a standard conformal mapping from the disc to the upper half plane and is essentially found in [10] , for instance. Let PW be the Paley-Wiener space of those $L^2\left(ℝ\right)$ functions supported on a compact subset of which Fourier transforms are possible, chosen to be $I=\left[-\pi ,\pi \right]$ but similar conclusions are valid for any compact interval. PW is equal to $PW={\mathcal{F}}^{-1}\left(L^2\left(I\right)\right)$ . These functions, as is well known, extend to complete functions over $ℂ$ of exponential type up to π in [11] . Recall that, given $f\in L^2\left(ℝ\right)$ ,

$P_{PW}f={\mathcal{F}}^{-1}\left({\chi }_I\hat{f}\right).$ (2)

Let $T_{\varphi }$ is Toeplitz operator on PW with respect to $\varphi$ so

$T_{\varphi }:PW\to PW,T_{\varphi }f=P_{PW}\left(\varphi f\right).$

Then,

${\left(T_{\varphi }f\right)}^{\wedge }={\chi }_I{\left(\varphi f\right)}^{\wedge }.$ (3)

Let $\text{sinc}\left(t\right)=\frac{\sin \pi t}{\pi t}$ (for $t\ne 0$ , $\text{sinc}\left(0\right)=1$ ). It is well-known that

$\text{sinc}\left(t-w\right)={\mathcal{F}}^{-1}\left(\frac{{\chi }_I\left(x\right){\text{e}}^{-iwx}}{\sqrt{2\pi }}\right).$ (4)

PW is a replicating kernel Hilbert space, as is also widely known [11] , with kernels denoted by $r_w\left(t\right)=\text{sinc}\left(t-\bar{w}\right)$ . Let any $f\in PW$ , $w\in ℂ$ ,

$f\left(w\right)=\langle f,r_w\rangle .$ (5)

3. Dividing the Symbol

In [12] , the Rochbergs technique is used in our analysis of Toeplitz operators $T_{\varphi }$ on PW. Pick and change some ${\psi }_L\in C_0^{\infty }\left(ℝ\right)$ ( $ℝ$ -based, indefinitely differentiable functions which tend to 0 at $\pm \infty$ ) so that support is given of ${\psi }_L$ lies in

$\left[-4\pi ,-\frac{\pi }{2}\right]$ , ${\psi }_L\equiv 1$ on $\left[-3\pi ,-\pi \right]$ and $0\le \psi \left(x\right)\le 1$ . Let ${\psi }_R\left(x\right)={\psi }_L(\; -\; x\; )$

and ${\psi }_C\equiv 0$ off I, ${\psi }_C\left(x\right)=1-{\psi }_L\left(x\right)-{\psi }_R\left(x\right)$ on I. Initially, we will assume that the symbols of our Toeplitz operators are in $W{L}^2\left(ℝ\right)$ , where W was previously defined. For this $\varphi$ , consider:

${\varphi }_L=\varphi \ast {\psi }_L,ˇ{\varphi }_C=\varphi \ast {\psi }_C,ˇ{\phi }_R=\varphi \ast {\psi }_R,ˇ$

where $\ast$ denotes convolution. Note that ${\psi }_R,{\psi }_C$ and ${\psi }_L$ all belong to $\mathcal{S}$ , the functions that experience rapid descent at infinity in the Schwartz space, in [13] for instance. Since the Fourier transform is a bijection on $\mathcal{S}$ , the Fourier transforms of these functions also belong to $\mathcal{S}$ . The next lemma demonstrates that each of ${\varphi }_L,{\varphi }_C$ and ${\varphi }_R$ is also a member of $W{L}^2\left(ℝ\right)$ .

Lemma (3.1): Suppose $\varphi \in W{L}^2\left(ℝ\right)$ and $\Delta \in \mathcal{S}$ then $\varphi \ast \Delta \in W{L}^2\left(ℝ\right)$ .

Proof: Let $f\in L^2\left(ℝ\right)$ . Then

$\langle \left(\varphi \ast \Delta \right)W^{-1},f\rangle ={\displaystyle {\int }_{ℝ}\Delta \left(s\right)}{\displaystyle {\int }_{ℝ}\varphi }\left(t-s\right)W^{-1}\left(t\right)\overline{f\left(t\right)}\text{d}t\text{d}s,$

based on the Tonelli and Fubini Theorems. A straightforward justification using the Cauchy-Schwarz inequality demonstrates that

$\left|{\displaystyle {\int }_{ℝ}\varphi }\left(t-s\right)W^{-1}\left(t\right)\overline{f\left(t\right)}\text{d}t\right|\le \Vert f\Vert \Vert \varphi W^{-1}\Vert \underset{t\in ℝ}{\sup }\left|\frac{W\left(t\right)}{W\left(t+s\right)}\right|.$

by way of a simple argument, ${\left|W\left(t\right)/W\left(t+s\right)\right|}^2\le 2s^2+2$ . then,

$\left|\langle \left(\varphi \ast \Delta \right)W^{-1},f\rangle \right|\le {\displaystyle {\int }_{ℝ}\left|\Delta \left(s\right)\right|}{\left(2s^2+2\right)}^{1/2}\text{d}s\Vert f\Vert \Vert \varphi W^{-1}\Vert .$

Given that $\Delta \in \mathcal{S}$ , the integral in the aforementioned expression is unquestionably finite. Consequently, $\left(\varphi \ast \Delta \right)W^{-1}\in L^2\left(ℝ\right)$ , according to the Riesz Representation Theorem [14] . Note that only a distributional perspective may be used to broadly consider the Fourier transforms of, $\varphi ,{\varphi }_R,{\varphi }_C$ and ${\varphi }_L$ . Their supports also fit into this category. We definitely have

$\text{supp}\left({\hat{\varphi }}_L\right)\subseteq \left[-4\pi ,-\frac{\pi }{2}\right],\text{supp}\left({\hat{\varphi }}_C\right)\subseteq \left[-\pi ,\pi \right],\text{supp}\left({\hat{\varphi }}_R\right)\subseteq \left[\frac{\pi }{2},4\pi \right].$

Recall that the Toeplitz operator with the sign depends simply on $\hat{\varphi }$ being restricted to $\left[-2\pi ,2\pi \right]$ . Consequently

$T_{\varphi }=T_{{\varphi }_L}+T_{{\varphi }_C}+T_{{\varphi }_R}.$ (6)

Additionally, the following, which is essentially illustrated in [12] , demonstrates that the symbol’s splitting is constant in the norm.

Lemma (3.2): Suppose ${\varphi }_X$ be ${\varphi }_L,{\varphi }_C$ or ${\varphi }_R$ . Then $K_X$ , a universal constant, exists in such a way that $\Vert T_{{\varphi }_X}f\Vert \le K_X\Vert T_{\varphi }f\Vert$ for all $f\in PW$ .

Proof: For $y\in ℝ$ , let $U_y$ be the translator specified by $U_{y}f\left(t\right)=f\left(t-y\right)$ . Then (in [12] p. 201), $T_{U_y}\varphi =U_y{T}_{\varphi }{U}_{-y}$ , so for all y, $T_{U_y\varphi }$ is unitarily equivalent to $T_{\varphi }$ . Let ${\text{Π}}_X={\text{Ψ}}_X^{ˇ}$ so we know that ${\varphi }_X=\varphi \ast {\Pi }_X$ and ${\Pi }_X\in \mathcal{S}\subseteq L^1\left(ℝ\right)$ . For any $g\in L^2\left(ℝ\right)$ and $t\in ℝ$ , $P_{PW}g\left(t\right)=\langle g,r_t\rangle$ by (5). This fact, along with the translation operator already described, reveals that $T_{{\varphi }_X}f\left(t\right)={\displaystyle {\int }_{ℝ}{T}_{U_y\varphi }f\left(t\right){\Pi }_X\left(y\right)\text{d}y}$ .

Therefore, by duality, we obtain

$\Vert T_{{\varphi }_X}f\Vert =\underset{g\in PW,\Vert g\Vert \le 1}{\sup }\left|\langle T_{{\varphi }_X}f,g\rangle \right|\le \underset{g\in PW,\Vert g\Vert \le 1}{\sup }\underset{y\in ℝ}{\sup }\left|\langle T_{U_y\varphi }f,g\rangle \right|{\Vert {\Pi }_X\Vert }_1={\Vert {\Pi }_X\Vert }_1\Vert T_{\varphi }f\Vert ,$

since for all $y\in ℝ$ , $T_{U_y\varphi }$ is unitarily equivalent to $T_{\varphi }$ . Therefore, for all $f\in PW$ ,

$\Vert T_{\varphi }f\Vert \approx \Vert T_{{\varphi }_L}f\Vert +\Vert T_{{\varphi }_C}f\Vert +\Vert T_{{\varphi }_R}f\Vert ,$ (7)

Specifically, universal constants $c,d$ exist such that

$c\left(\Vert T_{{\varphi }_L}f\Vert +\Vert T_{{\varphi }_C}f\Vert +\Vert T_{{\varphi }_R}f\Vert \right)\le \Vert T_{\varphi }f\Vert \le d\left(\Vert T_{{\varphi }_L}f\Vert +\Vert T_{{\varphi }_C}f\Vert +\Vert T_{{\varphi }_R}f\Vert \right).$

We will first test the boundedness and compactness of each of the three components before testing the boundedness and compactness of a Toeplitz operator. The action of multiplying by $\theta$ on $L^2\left(ℝ\right)$ is represented by $M_{\theta }$ , which $M_{\theta }$ is obviously a unitary operator if $\theta \left(t\right)={\text{e}}^{i\pi t}$ . The fact that, for example

$f\in L^2\left(ℝ\right)$ , ${\left(\theta f\right)}^{\wedge }\left(x\right)=\hat{f}\left(x-\pi \right)$ and ${\left(\bar{\theta }f\right)}^{\wedge }\left(x\right)=\hat{f}\left(x+\pi \right)$ .

The fact that follows holds for any $\psi \in W{L}^2\left(ℝ\right)$ and $f\in L^2\left(ℝ\right)$ , where at least one has a Fourier transform with compact support, will be used frequently throughout this paper:

$\text{supp}\left({\left(\psi f\right)}^{\wedge }\right)=\text{supp}\left(\hat{\psi }\ast \hat{f}\right)\subseteq \text{supp}\left(\hat{\psi }\right)+\text{supp}\left(\hat{f}\right),$ (8)

In [15] . In essence, [12] also contains the following lemma.

Lemma (3.3): Let $\psi \in W{L}^2\left(ℝ\right)$ , $\text{supp}\left(\hat{\psi }\right)\subseteq \left[\frac{\pi }{2},4\pi \right]$ . Then

$T_{\psi }=M_{\theta }{\text{Γ}}_{{\bar{\theta }}^2\psi }{M}_{\theta }$ and ${\text{Γ}}_{{\bar{\theta }}^2\psi }=M_{\bar{\theta }}{T}_{\psi }{P}_{PW}{M}_{\bar{\theta }}.$

Therefore, $\Vert T_{\psi }\Vert =\Vert {\Gamma }_{{\bar{\theta }}^2\psi }\Vert$ , provided these are finite, and $T_{\psi }$ is limited and closed only if ${\Gamma }_{{\bar{\theta }}^2\psi }$ is closed and bounded.

Proof: By taking into account the support provided by the Fourier transforms of pertinent functions, the first two equality conditions are established. Given that $M_{\theta }$ and PPW are bounded operators of norm 1, the operators’ norms are same, and their compactness is equivalent. For $w\in ℂ$ , let $h_w$ be the associated normalized reproducing kernel of PW.

$h_w\left(t\right)=\sqrt{\frac{2\pi \mathrm{Im}w}{\sinh \left(2\pi \mathrm{Im}w\right)}}\text{sinc}\left(t-\bar{w}\right),$ (9)

when $\mathrm{Im}w=0$ , with a suitable interpretation.

The component $T_{{\varphi }_R}$ boundedness will be established using the following statement.

Proposition (3.4): Let $\psi \in W{L}^2\left(ℝ\right)$ and $\text{supp}\left(\hat{\psi }\right)\subseteq \left[\frac{\pi }{2},4\pi \right]$ . Then $T_{\psi }$ is bounded if and only if it is bounded on $\left\{h_w:w\in {\mathcal{H}}_{+}\right\}$ . Additionally, a universal constant M exists such that $\Vert T_{\psi }\Vert \le M\sup \left\{\Vert T_{\psi }{h}_w\Vert :w\in {\mathcal{H}}_{+}\right\}$ .

Proof: Given that this is a collection of normalized functions, it is obvious that if $T_{\psi }$ is bounded, it is limited on $\left\{h_w:w\in {\mathcal{H}}_{+}\right\}$ . In contrast, $T_{\psi }$ is bounded by Lemma (2.1.4) if and only if ${\Gamma }_{{\bar{\theta }}^2\psi }$ is limited. According to Theorem (2.1.1), if ${\Gamma }_{{\bar{\theta }}^2\psi }$ is restricted on $\left\{j_w:w\in {\mathcal{H}}_{+}\right\}$ and such a universal constant M exists.

$\Vert {\Gamma }_{{\bar{\theta }}^2\psi }\Vert \le M\sup \left\{{\Gamma }_{{\bar{\theta }}^2\psi }{j}_w:w\in {\mathcal{H}}_{+}\right\}$ .

However, by Lemma (2.1.4), $\Vert {\Gamma }_{{\bar{\theta }}^2\psi }{j}_w\Vert =\Vert T_{\psi }{P}_{PW}{M}_{\bar{\theta }}{j}_w\Vert$ . A straightforward calculation shows that

$P_{PW}{M}_{\bar{\theta }}{j}_w\left(t\right)={\text{e}}^{-i\pi \bar{w}}\sqrt{2\pi \mathrm{Im}w}\text{sinc}\left(t-\bar{w}\right),$

and therefore

$\Vert {\Gamma }_{{\bar{\theta }}^2}\psi j_w\Vert ={\text{e}}^{-\pi \mathrm{Im}w}\sqrt{\sinh \left(2\pi \mathrm{Im}w\right)}\Vert T_{\psi }{h}_w\Vert .$ (10)

If ${\text{e}}^{-\pi \mathrm{Im}w}\sqrt{\sinh \left(2\pi \mathrm{Im}w\right)}<1$ for $w\in {\mathcal{H}}_{+}$ , where,

$\sup \left\{{\Gamma }_{{\bar{\theta }}^2\psi }{j}_w:w\in {\mathcal{H}}_{+}\right\}<\infty$ and so ${\Gamma }_{{\bar{\theta }}^2\psi }$

and hence $T_{\psi }$ is limited. Moreover,

$\Vert T_{\psi }\Vert =\Vert {\Gamma }_{{\bar{\theta }}^2\psi }\Vert \le M\sup \left\{{\Gamma }_{{\bar{\theta }}^2\psi }{j}_w:w\in {\mathcal{H}}_{+}\right\}\le M\sup \left\{\Vert T_{\psi }{h}_w\Vert :w\in {\mathcal{H}}_{+}\right\},$

as required. We will now talk about Toeplitz operators with symbols whose Fourier transforms are supported on $\left[-4\pi ,-\frac{\pi }{2}\right]$ , such as the component $T_{{\varphi }_L}$ .

We present the notation. $h\ast \left(t\right)=\overline{h\left(-t\right)}$ . Then it is simple to demonstrate that ${\left(\hat{h}\right)}^{\ast }={\left(\bar{h}\right)}^{\wedge }$ and that, for $g\in PW$ ,

${\left(T_{\bar{\psi }}g\right)}^{\wedge }={\left({\left(T_{\psi }\bar{g}\right)}^{\wedge }\right)}^{\ast }.$ (11)

Corollary (3.5): Let $\psi \in W{L}^2\left(ℝ\right)$ , $\text{supp}\left(\hat{\psi }\right)\subseteq \left[-4\pi ,-\frac{\pi }{2}\right]$ . $T_{\psi }$ is then said to be limited if and only if it has limits on $\left\{h_w:w\in {\mathcal{H}}_{-}\right\}$ . Additionally, An continuous constant M exists in a way that $\Vert T_{\psi }\Vert \le M\sup \left\{\Vert T_{\psi }{h}_w\Vert :w\in {\mathcal{H}}_{-}\right\}$ .

Proof: Given that complex conjugation, the Fourier transform, and the operations on $\ast$ are all unitary, $\Vert T_{\psi }\Vert =\Vert T_{\bar{\psi }}\Vert$ by (11). But we may apply Proposition (3.4) to $\psi$ . By (11) again, $\Vert T_{\bar{\psi }}{h}_w\Vert =\Vert T_{\psi }\overline{h_w}\Vert$ . It is easily shown that $\overline{h_w}=h_{\bar{w}}$ and the result therefore stands. In order to study the component $T_{{\varphi }_C}$ , we will now examine the scenario in which the symbol has support for the Fourier transform on $\left[-\pi ,\pi \right]$ .

Proposition (3.6): Let $\psi \in W{L}^2\left(ℝ\right)$ , $\hat{\psi }$ It is strengthened on $\left[-\pi ,\pi \right]$ and $w\in ℝ$ then $\psi \in L^{\infty }\left(ℝ\right)$ provided that $\sup \left\{T_{\psi }{h}_w:w\in ℝ\right\}<\infty$ , Additionally, there is a universal constant M such that $\sup {\Vert \psi \Vert }_{\infty }\le M\sup \left\{T_{\psi }{h}_w:w\in ℝ\right\}$ .

Proof: The inner product of $L^2\left(ℝ\right)$ functions and, more broadly, the impact of a distribution on a function are both shown here using the notation $\langle ,\rangle$ . According to the Paley-Wiener-Schwartz Theorem in [7] , $\psi$ is the limit to $ℝ$ of an entire function of exponential type at most π, as it has a compactly supported Fourier transform.

It is simple to demonstrate $\langle T_{\psi }{h}_w,h_w\rangle =\langle \hat{\psi },{\hat{h}}_w\ast {\hat{h}}_w\rangle$ and that

${\hat{h}}_w\ast h_w\left(x\right)=\frac{{\text{e}}^{-iwx}}{2\pi }\left(2\pi -\left|x\right|\right){\chi }_{2I}\left(x\right),$

using (4). Let $\Lambda \left(x\right)=\frac{2\pi -\left|x\right|}{\sqrt{2\pi }}{\chi }_{2I}\left(x\right)$ .

Then $\langle T_{\psi }{h}_w,h_w\rangle =\langle \hat{\psi },\Lambda {\hat{h}}_w\rangle =\langle \Lambda \hat{\psi },{\hat{h}}_w\rangle$ . However,

$\begin{array}{c}ˇ\Lambda \ast \psi \left(w\right)=\langle \Lambda \ast \psi ,{\delta }_w\rangle ˇ\text{where}{\delta }_w\text{isthepointmassat}w,\\ =\Lambda \hat{\psi },{\chi }_I{\hat{\delta }}_w,\text{as}\text{supp}\left(\hat{\psi }\right)\subseteq I,\\ =\Lambda \hat{\psi },{\hat{h}}_w.\end{array}$

Hence, $ˇ\Lambda \ast \psi \left(w\right)=\left\{T_{\psi }{h}_w,h_w\right\}$ and therefore,

$ˇ{\Vert \Lambda \ast \psi \Vert }_{\infty }=\underset{w\in ℝ}{\sup }\left|\langle T_{\psi }{h}_w,h_w\rangle \right|\le \underset{w\in ℝ}{\sup }\Vert T_{\psi }{h}_w\Vert ,$ (12)

as $\Vert h_w\Vert =1$ for all $w\in ℝ$ .

We aim to demonstrate this $ˇ{\Vert \Lambda \ast \psi \Vert }_{\infty }\approx {\Vert \psi \Vert }_{\infty }$ .

First off, we conclude that $ˇ\Lambda \in L^1\left(ℝ\right)$ as Λ is an array of ${\chi }_I\ast {\chi }_I$ , so $ˇ\Lambda$ is the square of an $L^2\left(ℝ\right)$ function. Therefore, $ˇ{\Vert \Lambda \ast \psi \Vert }_{\infty }\le {\Vert \Lambda \Vert }_1{\Vert \psi \Vert }_{\infty }ˇ$ . For the opposite inference, we see that $\psi \left(x\right)=\frac{\sqrt{\pi }}{2\pi -\left|x\right|}{\chi }_{2I}\left(x\right){\left(\Lambda \ast \psi \right)}^{\wedge }\left(x\right)ˇ$ .

On I, however, $ˇ{\left(\Lambda \ast \psi \right)}^{\wedge }$ is supported. As a result, we create a function V that, for $x\in I$ ,

$V\left(x\right)=\frac{\sqrt{2\pi }}{2\pi -\left|x\right|},$

After that, and V is expanded to become even, supported by 2I, and infinitely differentiable, save at 0. Therefore, since $\hat{\varphi }=V{\left(\Lambda \ast \varphi \right)}^{\wedge }ˇ$ so $ˇ\varphi =V\ast \left(\hat{\Lambda }\ast \varphi \right)$ The math below indicates that $ˇV\in L^1\left(ℝ\right)$ .

$\begin{array}{c}ˇV\left(t\right)=\sqrt{\frac{2}{\pi }}{\displaystyle {\int }_0^{2\pi }V\left(x\right)\cos \left(tx\right)\text{d}x},\text{as}V\text{iseven}\\ =-\sqrt{\frac{2}{\pi }}\frac{1}{t^2}\left({V^{\prime }}_{+}\left(0\right)+{\displaystyle {\int }_0^{2\pi }{V}^{″}\left(x\right)\cos \left(tx\right)\text{d}x}\right),\end{array}$

utilizing two integrations via sections. Therefore, $ˇ\left|V\left(t\right)\right|\le \frac{2}{\pi }\frac{1}{t^2}\left|{V^{\prime }}_{+}\left(0\right)\right|+\frac{1}{2}{\Vert V\Vert }_1$ .

Being continuous, $Vˇ$ is unquestionably locally integrable, hence $V\in L^1\left(ℝ\right)$ follows.

Hence, $ˇ{\Vert \psi \Vert }_{\infty }\le {\Vert V\Vert }_1{\Vert \Lambda \ast \psi \Vert }_{\infty }ˇ$ , so that $ˇ{\Vert \Lambda \ast \psi \Vert }_{\infty }\approx {\Vert \psi \Vert }_{\infty }$ . By combining this with (12), we arrive at the desired outcome.

4. Main Findings

The first fundamental theorem can now be stated. $T_{\varphi }$ is divided into $T_{{\varphi }_L}+T_{{\varphi }_C}+T_{{\varphi }_R}$ , We shall see that the boundedness of $T_{\varphi }$ on $\left\{h_w:w\in {\mathcal{H}}_{+}\right\}$ . determines the boundedness of $T_{{\varphi }_R}$ . The boundedness of $T_{\varphi }$ on $\left\{h_w:w\in {\mathcal{H}}_{-}\right\}$ and $\left\{h_w:w\in ℝ\right\}$ , respectively, determines the boundedness of $T_{{\varphi }_L}$ and $T_{{\varphi }_C}$ .

Theorem (4.1): Let $\varphi \in W{L}^2\left(ℝ\right)$ . Subsequently, $T_{\varphi }$ is limited if and only if it is limited on $\left\{h_w:w\in ℂ\right\}$ . Additionally, a continuous constant M exists in a way that

$\Vert T_{\varphi }\Vert \le M\sup \left\{\Vert T_{\varphi }{h}_w\Vert :w\in ℂ\right\}.$

Proof: Clearly, if $T_{\varphi }$ is bounded, it is bounded on $\left\{h_w:w\in ℂ\right\}$ since these are a collection of normalized functions. Conversely, we know by (7) that for all $f\in PW$ ,

$\Vert T_{\varphi }f\Vert \approx \Vert T_{{\varphi }_L}f\Vert +\Vert T_{{\varphi }_C}f\Vert +\Vert T_{{\varphi }_R}f\Vert .$ (13)

By Proposition (3.4), $T_{{\varphi }_R}$ is limited provided that $\sup \left\{T_{{\varphi }_R}{h}_w:w\in {\mathcal{H}}_{+}\right\}<\infty$ and this is undoubtedly accurate given that

$\sup \left\{\Vert T_{\varphi }{h}_w\Vert :w\in {\mathcal{H}}_{+}\right\}<\infty ,$ (14)

through Lemma (3.2). Similar to that, according to Corollary (3.5) and Lemma (3.2), $T_{{\varphi }_L}$ is limited if

$\sup \left\{\Vert T_{\varphi }{h}_w\Vert :w\in {\mathcal{H}}_{-}\right\}<\infty ,$ (15)

Rochberg shows that $\Vert T_{{\varphi }_C}\Vert \approx {\Vert {\varphi }_C\Vert }_{\infty }$ , in [12] . $T_{{\varphi }_C}$ is therefore constrained by Proposition (3.6) and Lemma (3.2), provided that

$\sup \left\{\Vert T_{\varphi }{h}_w\Vert :w\in ℝ\right\}<\infty .$ (16)

When (13), (14), (15), and (16) are combined, we discover that $T_{\varphi }$ is bounded if $\sup \left\{\Vert T_{\varphi }{h}_w\Vert :w\in ℂ\right\}<\infty$ . The estimate for $\Vert T_{\varphi }\Vert$ is similarly produced by estimating the norms of $T_{{\varphi }_L},T_{{\varphi }_R}$ and $T_{{\varphi }_R}$ using Proposition (3.4), Corollary 2.5, and Proposition (3.6). By using a counterexample, we can demonstrate that the supremum of the norms of the pictures in the set of $h_w$ for $w\in ℝ$ is not comparable to the norm of a Toeplitz operator.

Lemma (4.2): No universal constant M exists such that $\Vert T_{\varphi }\Vert \le M\sup \left\{\Vert T_{\varphi }{h}_w\Vert :w\in ℝ\right\}$ , for all bounded $T_{\varphi }$ .

Proof: For $\alpha \in \left[0,2\pi \right)$ , let ${\varphi }_{\alpha }\left(t\right)={\text{e}}^{i\alpha t}$ . Since ${\Vert {\varphi }_{\alpha }\Vert }_{\infty }=1$ it is clear that $\Vert T_{{\varphi }_{\alpha }}\Vert \le 1$ . Fourier transforms are used in an easy computation to demonstrate that

${\Vert T_{{\varphi }_{\alpha }}{h}_w\Vert }^2=\frac{{\text{e}}^{v\alpha }\sinh \left(2\pi -\alpha \right)v}{\sinh \left(2\pi v\right)}.$

where $v=\mathrm{Im}w$ (if v = 0, given a reasonable interpretation). In particular, if $w\in ℝ$ then

${\Vert T_{{\varphi }_{\alpha }}{h}_w\Vert }^2=\frac{2\pi -\alpha }{2\pi }\text{so}\sup \left\{\Vert T_{{\varphi }_{\alpha }}{h}_w\Vert :w\in ℝ\right\}=\sqrt{\frac{2\pi -\alpha }{2\pi }}.$ (17)

However, ${\lim }_{v\to \infty }\frac{{\text{e}}^{v\alpha }\sinh \left(2\pi -\alpha \right)v}{\sinh \left(2\pi v\right)}=1$ .

Therefore, $\sup \left\{T_{{\varphi }_{\alpha }}{h}_w:w\in ℂ\right\}$ 1, so $\Vert T_{{\varphi }_{\alpha }}\Vert =1$ . Therefore, any such M would need satisfy $M\ge \sqrt{\frac{2\pi }{2\pi -\alpha }}$ , for all values of $\alpha \in \left[0,2\pi \right)$ , which is obviously not possible. We will start by demonstrating a statement that is true for any compact operator on PW.

Proposition (4.3): If T is any compact operator from PW to a Hilbert space H, assumable, ${\lim }_{\left|w\right|\to \infty }\Vert T{h}_w\Vert =0$ .

Proof: To do this, we must first demonstrate that $h_w$ weakly converges to zero as

$\left|w\right|\to \infty$ . $E={\displaystyle {\cup }_{\delta \in \left(0,\pi \right)}{\mathcal{F}}^{-1}{L}^2}\left(\left[-\pi +\delta ,\pi -\delta \right]\right)$ .

It is then simple to demonstrate that E is a dense subspace of PW. Let $f\in E$ . Using the kernels’ ability to reproduce,

$\langle f,h_w\rangle =f\left(w\right)\sqrt{\frac{2\pi v}{\sinh \left(2\pi v\right)}},$ (18)

where $v=\mathrm{Im}w$ , by (5) and (9). There is a constant $K_f$ such that for some $\pi -\delta$ , $\delta \in \left(0,\pi \right)$ , f is a complete function of exponential type at most.

$\left|f\left(w\right)\right|\le K_f{\text{e}}^{\left(\pi -\delta \right)\left|v\right|}$ , in [11] . Therefore,

$\left|f,h_w\right|\le K_f{\text{e}}^{\left(\pi -\delta \right)\left|v\right|}\sqrt{\frac{2\pi v}{\sinh \left(2\pi v\right)}}=\frac{2K_f{\text{e}}^{-\delta \left|v\right|}\sqrt{\pi \left|v\right|}}{\sqrt{1-{\text{e}}^{-4\pi \left|v\right|}}}\to 0\text{as}\left|v\right|\to \infty .$

We can also show that, for any $R>0$ , $\left|f\left(u+iv\right)\right|\to 0$ as $\left|u\right|\to \infty$ , this is a generalization of the Riemann-Lebesgue Lemma for $\left|v\right|\le R^{-}$ . Using (18) once more, it is evident that ${\lim }_{\left|w\right|\to \infty }\langle f,h_w\rangle =0$ .

However, a typical argument demonstrates that $h_w$ converges weakly to zero as $\left|w\right|\to \infty$ .| since E is a dense subspace of PW and $\left\{h_w:w\in ℂ\right\}$ is uniformly bounded. Compact operators transform weak convergence to norm convergence, so the outcome is as shown in [16] . If T is a Toeplitz operator, we can get the opposite of this conclusion.

Theorem (4.4): Let $\varphi \in W{L}^2\left(ℝ\right)$ . Then $T_{\varphi }$ is closed and bounded if and only if

$\underset{\left|w\right|\to \infty }{\lim }\Vert T_{\varphi }{h}_w\Vert =0.$

Proof: The prior assertion provides the forward implication. Note that by (7) $T_{\varphi }$ is compact if and only if each of $T_{{\varphi }_L},T_{{\varphi }_C}$ , and $T_{{\varphi }_R}$ ) is, demonstrating the opposite conclusion. Let’s start by thinking about $T_{{\varphi }_R}$ . According to our theory and (7),

$\underset{w\in {\mathcal{H}}_{+},\left|w\right|\to \infty }{\lim }\Vert T_{{\varphi }_R}{h}_w\Vert =0.$

By (10),

$\Vert {\Gamma }_{{\bar{\theta }}^2}{\varphi }_R{j}_w\Vert ={\text{e}}^{-\pi \mathrm{Im}w}\sqrt{\sinh \left(2\pi \mathrm{Im}w\right)}\Vert T_{{\varphi }_R}{h}_w\Vert .$

Therefore,

$\underset{w\in {\mathcal{H}}_{+},\left|w\right|\to \infty }{\lim }\Vert {\Gamma }_{{\bar{\theta }}^2{\varphi }_R}{j}_w\Vert =\underset{w\in {\mathcal{H}}_{+},\mathrm{Im}w\to 0}{\lim }\Vert {\Gamma }_{{\bar{\theta }}^2{\varphi }_R}{j}_w\Vert =0.$

As a result, according to Theorem (2.1), ${\Gamma }_{{\bar{\theta }}^2{\varphi }_R}$ is compact, and by Lemma (3.3), $T_{{\varphi }_R}$ is closed and bounded. The same argument (applied to $T_{\overline{{\varphi }_L}}$ and using (11)) shows that $T_{\overline{{\varphi }_L}}$ and hence $T_{{\varphi }_L}$ is compact. $T_{{\varphi }_C}$ is compact provided that $\left|{\varphi }_C\left(w\right)\right|\to 0$ as $\left|w\right|\to \infty$ , in [12] . $T_{{\varphi }_C}$ is therefore implied to be closed and bounded by Lemma (3.2) and Proposition (3.6), according to our hypothesis. $T_{\varphi }$ is hence compact.

5. Other Issues

It is possible to determine whether Toeplitz operators belong to a certain Schatten-von Neumann class by observing how the operators behave on the reproducing kernels. It would be interesting to see if similar results are obtained for Hankel-type operators on PW. For this space, Hankel-type operators in one form are considered in [12] , and it is found that they are equivalent to the Toeplitz operators considered in this paper. However, Hankel-type operators defined on PW by

$H_{\varphi }:PW\to P{W}^{\perp },H_{\varphi }f=P_{P{W}^{\perp }}\left(\varphi f\right),$

do not appear to have been analyses.

Acknowledgements

This study is supported via funding from Prince Sattam bin Abdulaziz University Project number (PSAU/2023/R/144).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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