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
is orthogonal projection onto E,
is orthogonal complement of E for E, a closed subspace of a Hilbert space. Let
be the closed subspace of
containing those functions that vanish, or are outside of J, for J, a measurable subset of
, the real line, and
convey the characteristic function of J. We define in
) inner product by
, as well as the
function and operator norms by
. Other
norms will be identified by the symbol
. The (unitary) Fourier transform and its inverse will be denoted by the following definitions and notations:
2. Hankel Operators in the Hardy Space
Suppose
and
are lower and half upper planes of
. We define
is the Hardy space of the upper half plane. By the theorem of Paley–Wiener,
, in [8] . Therefore, for
,
(1)
If
. For
, let
is a Hankel operator on
with symbol
so:
See [9] for more information on these operators. It is fundamental that
for
to be defined, since
. Note that under this condition,
is at least defined on
, a dense subspace of
, also:
.
The following theorem provides both essential and enough criteria for
to be bounded and closed (also known as “Theorem of Bonsall’s”).
Theorem (2.1): For
, let
be
normalized Replication of kernel connected to w, that is
Then, for
,
is limited if and only if it has limits on every
. Additionally, there is a universal constant M that exists such that
In addition,
is compact if and only if
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
functions supported on a compact subset of which Fourier transforms are possible, chosen to be
but similar conclusions are valid for any compact interval. PW is equal to
. These functions, as is well known, extend to complete functions over
of exponential type up to π in [11] . Recall that, given
,
(2)
Let
is Toeplitz operator on PW with respect to
so
Then,
(3)
Let
(for
,
). It is well-known that
(4)
PW is a replicating kernel Hilbert space, as is also widely known [11] , with kernels denoted by
. Let any
,
,
(5)
3. Dividing the Symbol
In [12] , the Rochbergs technique is used in our analysis of Toeplitz operators
on PW. Pick and change some
(
-based, indefinitely differentiable functions which tend to 0 at
) so that support is given of
lies in
,
on
and
. Let
and
off I,
on I. Initially, we will assume that the symbols of our Toeplitz operators are in
, where W was previously defined. For this
, consider:
where
denotes convolution. Note that
and
all belong to
, the functions that experience rapid descent at infinity in the Schwartz space, in [13] for instance. Since the Fourier transform is a bijection on
, the Fourier transforms of these functions also belong to
. The next lemma demonstrates that each of
and
is also a member of
.
Lemma (3.1): Suppose
and
then
.
Proof: Let
. Then
based on the Tonelli and Fubini Theorems. A straightforward justification using the Cauchy-Schwarz inequality demonstrates that
by way of a simple argument,
. then,
Given that
, the integral in the aforementioned expression is unquestionably finite. Consequently,
, according to the Riesz Representation Theorem [14] . Note that only a distributional perspective may be used to broadly consider the Fourier transforms of,
and
. Their supports also fit into this category. We definitely have
Recall that the Toeplitz operator with the sign depends simply on
being restricted to
. Consequently
(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
be
or
. Then
, a universal constant, exists in such a way that
for all
.
Proof: For
, let
be the translator specified by
. Then (in [12] p. 201),
, so for all y,
is unitarily equivalent to
. Let
so we know that
and
. For any
and
,
by (5). This fact, along with the translation operator already described, reveals that
.
Therefore, by duality, we obtain
since for all
,
is unitarily equivalent to
. Therefore, for all
,
(7)
Specifically, universal constants
exist such that
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
on
is represented by
, which
is obviously a unitary operator if
. The fact that, for example
,
and
.
The fact that follows holds for any
and
, where at least one has a Fourier transform with compact support, will be used frequently throughout this paper:
(8)
In [15] . In essence, [12] also contains the following lemma.
Lemma (3.3): Let
,
. Then
and
Therefore,
, provided these are finite, and
is limited and closed only if
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
and PPW are bounded operators of norm 1, the operators’ norms are same, and their compactness is equivalent. For
, let
be the associated normalized reproducing kernel of PW.
(9)
when
, with a suitable interpretation.
The component
boundedness will be established using the following statement.
Proposition (3.4): Let
and
. Then
is bounded if and only if it is bounded on
. Additionally, a universal constant M exists such that
.
Proof: Given that this is a collection of normalized functions, it is obvious that if
is bounded, it is limited on
. In contrast,
is bounded by Lemma (2.1.4) if and only if
is limited. According to Theorem (2.1.1), if
is restricted on
and such a universal constant M exists.
.
However, by Lemma (2.1.4),
. A straightforward calculation shows that
and therefore
(10)
If
for
, where,
and so
and hence
is limited. Moreover,
as required. We will now talk about Toeplitz operators with symbols whose Fourier transforms are supported on
, such as the component
.
We present the notation.
. Then it is simple to demonstrate that
and that, for
,
(11)
Corollary (3.5): Let
,
.
is then said to be limited if and only if it has limits on
. Additionally, An continuous constant M exists in a way that
.
Proof: Given that complex conjugation, the Fourier transform, and the operations on
are all unitary,
by (11). But we may apply Proposition (3.4) to
. By (11) again,
. It is easily shown that
and the result therefore stands. In order to study the component
, we will now examine the scenario in which the symbol has support for the Fourier transform on
.
Proposition (3.6): Let
,
It is strengthened on
and
then
provided that
, Additionally, there is a universal constant M such that
.
Proof: The inner product of
functions and, more broadly, the impact of a distribution on a function are both shown here using the notation
. According to the Paley-Wiener-Schwartz Theorem in [7] ,
is the limit to
of an entire function of exponential type at most π, as it has a compactly supported Fourier transform.
It is simple to demonstrate
and that
using (4). Let
.
Then
. However,
Hence,
and therefore,
(12)
as
for all
.
We aim to demonstrate this
.
First off, we conclude that
as Λ is an array of
, so
is the square of an
function. Therefore,
. For the opposite inference, we see that
.
On I, however,
is supported. As a result, we create a function V that, for
,
After that, and V is expanded to become even, supported by 2I, and infinitely differentiable, save at 0. Therefore, since
so
The math below indicates that
.
utilizing two integrations via sections. Therefore,
.
Being continuous,
is unquestionably locally integrable, hence
follows.
Hence,
, so that
. By combining this with (12), we arrive at the desired outcome.
4. Main Findings
The first fundamental theorem can now be stated.
is divided into
, We shall see that the boundedness of
on
. determines the boundedness of
. The boundedness of
on
and
, respectively, determines the boundedness of
and
.
Theorem (4.1): Let
. Subsequently,
is limited if and only if it is limited on
. Additionally, a continuous constant M exists in a way that
Proof: Clearly, if
is bounded, it is bounded on
since these are a collection of normalized functions. Conversely, we know by (7) that for all
,
(13)
By Proposition (3.4),
is limited provided that
and this is undoubtedly accurate given that
(14)
through Lemma (3.2). Similar to that, according to Corollary (3.5) and Lemma (3.2),
is limited if
(15)
Rochberg shows that
, in [12] .
is therefore constrained by Proposition (3.6) and Lemma (3.2), provided that
(16)
When (13), (14), (15), and (16) are combined, we discover that
is bounded if
. The estimate for
is similarly produced by estimating the norms of
and
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
for
is not comparable to the norm of a Toeplitz operator.
Lemma (4.2): No universal constant M exists such that
, for all bounded
.
Proof: For
, let
. Since
it is clear that
. Fourier transforms are used in an easy computation to demonstrate that
where
(if v = 0, given a reasonable interpretation). In particular, if
then
(17)
However,
.
Therefore,
1, so
. Therefore, any such M would need satisfy
, for all values of
, 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,
.
Proof: To do this, we must first demonstrate that
weakly converges to zero as
.
.
It is then simple to demonstrate that E is a dense subspace of PW. Let
. Using the kernels’ ability to reproduce,
(18)
where
, by (5) and (9). There is a constant
such that for some
,
, f is a complete function of exponential type at most.
, in [11] . Therefore,
We can also show that, for any
,
as
, this is a generalization of the Riemann-Lebesgue Lemma for
. Using (18) once more, it is evident that
.
However, a typical argument demonstrates that
converges weakly to zero as
.| since E is a dense subspace of PW and
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
. Then
is closed and bounded if and only if
Proof: The prior assertion provides the forward implication. Note that by (7)
is compact if and only if each of
, and
) is, demonstrating the opposite conclusion. Let’s start by thinking about
. According to our theory and (7),
By (10),
Therefore,
As a result, according to Theorem (2.1),
is compact, and by Lemma (3.3),
is closed and bounded. The same argument (applied to
and using (11)) shows that
and hence
is compact.
is compact provided that
as
, in [12] .
is therefore implied to be closed and bounded by Lemma (3.2) and Proposition (3.6), according to our hypothesis.
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
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).