Composition Operators in Paley–Wiener Spaces

• Composition operators in Paley–Wiener spaces are defined by f∘φ and exhibit affine rigidity, meaning only affine maps (with c in ℝ and 0<|c|≤1) yield bounded operators.
• The analysis employs reproducing kernel Hilbert space techniques and Fourier theory to precisely characterize norms, spectral properties, and dynamic behaviors such as cyclicity and expansivity.
• Extensions to range spaces and weighted compositions reinforce that the operator structure, spectrum, and dynamical properties are strictly governed by the underlying affine geometry.

Composition operators on Paley–Wiener spaces are operators of the form $C_\phi f=f\circ\phi$ acting on spaces of bandlimited entire functions. The Paley–Wiener theorem identifies these spaces with subspaces of entire functions of exponential type, and in the standard notation $B^2_a$ or $PW_a$ they consist of entire functions of exponential type at most $a$ whose restrictions to $\mathbb R$ belong to $L^2(\mathbb R)$, equivalently functions whose Fourier transform is supported in $[-a,a]$ (Mukherjee et al., 2010). In this setting the operator theory is unusually rigid: boundedness forces the symbol to be affine, and recent work gives explicit classifications of compactness, spectrum, spectral radius, cyclicity, Li–Yorke chaos, positive expansivity, positive shadowing property, and absolutely Cesàro boundedness (Álvarez et al., 27 Aug 2025).

1. Functional setting and affine rigidity

The Paley–Wiener space $B^2_a$ is a reproducing kernel Hilbert space with kernel

$$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$

and the notation $B^2_\sigma$ is also used when the bandwidth parameter is written as $B^2_a$0 (Mukherjee et al., 2010). A composition operator is defined by

$B^2_a$1

For Paley–Wiener spaces, the bounded symbols are completely characterized. A composition operator maps $B^2_a$2 boundedly into itself if and only if

$B^2_a$3

for some $B^2_a$4, $B^2_a$5, and $B^2_a$6 (Mukherjee et al., 2010). The same characterization appears in the later $B^2_a$7 analysis, where the only bounded composition operators are those induced by affine mappings

$B^2_a$8

(Álvarez et al., 27 Aug 2025).

This classification expresses a strong rigidity phenomenon. Nonlinear symbols, even when analytic or entire, do not preserve Paley–Wiener spaces. The result is sometimes described as invariance only under real affine maps with non-expanding real linear part, and it underlies essentially all subsequent spectral and dynamical classifications (Mukherjee et al., 2010).

2. Reproducing-kernel and Fourier-theoretic framework

A broader RKHS framework places the Paley–Wiener case inside a larger class of reproducing kernel Hilbert spaces associated with analytic positive definite functions. In that setting one considers

$B^2_a$9

where $PW_a$0 is non-negative almost everywhere. Under the assumptions

$PW_a$1

and

$PW_a$2

spans the space of real $PW_a$3 matrices, bounded composition operators are again induced only by affine maps $PW_a$4 with $PW_a$5 and $PW_a$6 (Ikeda et al., 2019).

The analysis is formulated intrinsically in RKHS language. The feature map is

$PW_a$7

and there is an explicit isomorphism

$PW_a$8

with norm

$PW_a$9

(Ikeda et al., 2019). In this formulation, boundedness of $a$0 is reduced to boundedness of an operator on polynomials in weighted $a$1-spaces.

A notable aspect of the general theory is that it does not require finite order of the associated entire functions. The method relies on intrinsic properties of the RKHSs and on asymptotic properties of the greatest zeros of orthogonal polynomials on weighted $a$2-spaces on the real line. The paper gives the example

$a$3

for which $a$4 is an entire function of infinite order, yet only affine maps induce bounded composition operators (Ikeda et al., 2019). This suggests that the affine rigidity is a structural consequence of the RKHS/Fourier geometry rather than an artifact of finite-order entire-function theory.

3. Operator structure: adjoint, invertibility, norm, compactness, and closed range

For bounded composition operators on $a$5, the adjoint acts simply on reproducing kernels: $a$6 (Álvarez et al., 27 Aug 2025). This kernel formula is central in the compactness analysis and in several orbit arguments.

Invertibility is also explicit. If $a$7, then $a$8 is a bounded invertible operator if and only if $a$9 (Álvarez et al., 27 Aug 2025). The cases $\mathbb R$0 are therefore bounded but non-invertible.

The operator norm admits a sharp two-sided estimate. For $\mathbb R$1, the operator $\mathbb R$2 is an isometry on $\mathbb R$3, and for general $\mathbb R$4,

$\mathbb R$5

with equality when $\mathbb R$6 (Álvarez et al., 27 Aug 2025). The dependence on $\mathbb R$7 already anticipates the distinction between real and non-real translations in the spectral and dynamical theory.

Compactness fails completely. There are no compact composition operators on any Paley–Wiener space $\mathbb R$8 (Álvarez et al., 27 Aug 2025). In the $\mathbb R$9 proof this is shown by reproducing kernel estimates: weak convergence of normalized kernels together with the lack of norm convergence of their images under $L^2(\mathbb R)$0 excludes compactness, using kernels $L^2(\mathbb R)$1. In the broader RKHS setting the same conclusion is obtained from translation-invariant kernel structure and the Riemann–Lebesgue theorem; the resulting contrast with other entire-function spaces, such as Fock spaces, is explicit (Ikeda et al., 2019).

Another rigid property is closed range: every bounded composition operator on $L^2(\mathbb R)$2 has closed range, a fact obtained via similarity and the preservation of the closed-range property under similarity (Álvarez et al., 27 Aug 2025).

4. Spectrum and spectral radius

For $L^2(\mathbb R)$3 with $L^2(\mathbb R)$4, $L^2(\mathbb R)$5, and $L^2(\mathbb R)$6, the spectrum is completely characterized (Álvarez et al., 27 Aug 2025).

Symbol class	Spectrum $L^2(\mathbb R)$7	Spectral radius $L^2(\mathbb R)$8
$L^2(\mathbb R)$9	$[-a,a]$0	$[-a,a]$1
$[-a,a]$2	$[-a,a]$3	$[-a,a]$4
$[-a,a]$5	$[-a,a]$6	$[-a,a]$7

The spectral radius formula is

$[-a,a]$8

and the corresponding spectrum is

$[-a,a]$9

For $B^2_a$0, the key method is that the operator is similar to an isometry, and when $B^2_a$1 it is non-invertible. For $B^2_a$2, the operator corresponds to a multiplication operator on $B^2_a$3 (Álvarez et al., 27 Aug 2025).

The $B^2_a$4 case is especially transparent: translations in the complex plane become multiplication by $B^2_a$5 in the Fourier model. This is the mechanism behind both the spectral description and the later cyclicity criteria.

5. Linear dynamics on $B^2_a$6

The linear-dynamical behavior of bounded composition operators on $B^2_a$7 is fully classified. For Li–Yorke chaos, the relevant notion is the existence of an irregular vector $B^2_a$8 satisfying

$B^2_a$9

No bounded composition operator on $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$0 is Li–Yorke chaotic (Álvarez et al., 27 Aug 2025). The proof splits by parameter regime: for $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$1, orbit norms grow at least as $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$2 for some $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$3, so minimal norms do not approach zero; for $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$4, the operators are normal; for $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$5, norms remain bounded.

Positive expansivity behaves differently. A bounded composition operator is positively expansive if and only if

$$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$6

(Álvarez et al., 27 Aug 2025). In the first case norms of orbits tend to infinity; in the second they grow exponentially, asymptotically like $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$7.

The positive shadowing property never occurs. No bounded composition operator on $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$8 has the positive shadowing property (Álvarez et al., 27 Aug 2025). For $$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi(z-\overline{w})},$$9, the obstruction is spectral: for normal operators this property is equivalent to $B^2_\sigma$0, which fails because the spectrum always meets the unit circle. For $B^2_\sigma$1, counterexamples are constructed using fixed points of $B^2_\sigma$2.

Absolute Cesàro boundedness is also explicit. The defining condition is the existence of $B^2_\sigma$3 such that

$B^2_\sigma$4

For composition operators on $B^2_\sigma$5,

$B^2_\sigma$6

In these cases orbits are uniformly bounded, whereas for $B^2_\sigma$7 and for $B^2_\sigma$8, the Cesàro means are unbounded because of rapid norm growth (Álvarez et al., 27 Aug 2025).

A central structural point is that positive expansivity, Li–Yorke chaos, and absolute Cesàro boundedness separate cleanly in this setting. Orbit growth alone does not produce chaotic behavior, and spectral normality suppresses several forms of instability.

6. Cyclicity, adjoints, supercyclicity, and complex symmetry

The cyclicity problem for bounded composition operators on Paley–Wiener spaces $B^2_\sigma$9 is also solved completely. Writing the symbol in the form

$B^2_a$00

$B^2_a$01 is cyclic precisely when

$B^2_a$02

and either $B^2_a$03 or $B^2_a$04 with

$B^2_a$05

(Hai et al., 2024). Thus cyclicity occurs only for translations, and among real translations only for the specified size range.

No composition operator $B^2_a$06 on any $B^2_a$07 is supercyclic (Hai et al., 2024). The non-cyclic cases are also explicit: if $B^2_a$08, then $B^2_a$09 is never cyclic; if $B^2_a$10, it is never cyclic because the orbit has at most two elements.

The Fourier representation explains the translation case. When $B^2_a$11, $B^2_a$12 is normal and corresponds to multiplication by $B^2_a$13 on $B^2_a$14. By spectral theory, this multiplication operator is cyclic if and only if the multiplier is injective on a set of full measure. Hence cyclicity always holds when $B^2_a$15, while for real $B^2_a$16 it holds exactly when the period $B^2_a$17 is at least the interval length $B^2_a$18, equivalently $B^2_a$19 (Hai et al., 2024).

Adjoints display a different pattern. For $B^2_a$20, $B^2_a$21 is cyclic, and in fact any reproducing kernel is a cyclic vector. More generally,

$B^2_a$22

(Hai et al., 2024).

The cyclicity of reproducing kernels is tied to completeness of exponential systems. For $B^2_a$23, every kernel $B^2_a$24 is cyclic for $B^2_a$25 if $B^2_a$26 or $B^2_a$27 with $B^2_a$28; if $B^2_a$29, no kernel is cyclic (Hai et al., 2024). In the Fourier picture, $B^2_a$30 corresponds to $B^2_a$31, and the orbit corresponds to $B^2_a$32. Density is therefore equivalent to completeness of $B^2_a$33 in $B^2_a$34, and Carleman’s theorem gives completeness exactly for $B^2_a$35.

Complex symmetry provides another structural layer. $B^2_a$36 is complex symmetric exactly when either $B^2_a$37, $B^2_a$38, or $B^2_a$39, $B^2_a$40; and $B^2_a$41 is normal exactly when $B^2_a$42, $B^2_a$43, or $B^2_a$44, $B^2_a$45 (Hai et al., 2024). When complex symmetry holds, cyclicity of $B^2_a$46 and $B^2_a$47 coincide.

7. Range spaces, weighted composition, and extensions beyond invariance

Although only affine symbols preserve a Paley–Wiener space, arbitrary composition can still be analyzed through the range of $B^2_a$48. For suitable entire maps $B^2_a$49, the range $B^2_a$50 is a reproducing kernel Hilbert space with kernel

$B^2_a$51

(Mukherjee et al., 2010). If $B^2_a$52, then

$B^2_a$53

so $B^2_a$54 is an isometry onto its range, and the functions

$B^2_a$55

form an orthonormal basis of $B^2_a$56 (Mukherjee et al., 2010).

For measurable $B^2_a$57, boundedness of $B^2_a$58 on $B^2_a$59 is characterized by the existence of $B^2_a$60 such that

$B^2_a$61

for all measurable $B^2_a$62, where $B^2_a$63 is Lebesgue measure (Mukherjee et al., 2010). Under this criterion, for any $B^2_a$64 and any $B^2_a$65, there exists $B^2_a$66 for large enough $B^2_a$67 such that

$B^2_a$68

Thus the image of a bandlimited function under a more general warp need not remain bandlimited, but it can still be approximated arbitrarily well inside a larger Paley–Wiener space.

Weighted composition extends the picture further. If one allows multiplication by an entire function $B^2_a$69, with $B^2_a$70 and $B^2_a$71, then

$B^2_a$72

where

$B^2_a$73

(Mukherjee et al., 2010).

The same rigidity persists in de Branges–Rovnyak spaces $B^2_a$74, with Paley–Wiener spaces as the special case $B^2_a$75. If $B^2_a$76 is entire, of exponential type $B^2_a$77, and satisfies $B^2_a$78 in the upper half-plane, then a composition operator $B^2_a$79 is bounded on $B^2_a$80 only if $B^2_a$81 is affine, $B^2_a$82, with $B^2_a$83, $B^2_a$84, and $B^2_a$85 (Mukherjee et al., 2010). This broader perspective reinforces the central conclusion: Paley–Wiener composition theory is governed by affine geometry, Fourier support, and RKHS structure, with explicit operator-theoretic consequences across spectral theory, dynamics, and range analysis.