Spectrum of composition operators on $\mathcal{S}(\mathbb{R})$ with polynomial symbols

Abstract

We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to {0}, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origin. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.

Introduction

Composition operators on Fréchet spaces of smooth functions on the reals have attracted the attention of several authors recently ([7], [8], [12], [16], [17], [18], [19]) but to our knowledge very little is known about the spectra of composition operators in this setting. See for instance [3] or [5], where the spectrum of composition operators and other classical operators on spaces of real smooth functions is investigated. This contrasts with the large number of existing articles studying spectral properties of composition operators in Banach spaces of analytic functions on the unit disc.

We study the spectrum of composition operators defined in the Schwartz space $\mathcal{S}(\mathbb{R})$ of smooth rapidly decreasing functions, $C_{\phi }:\mathcal{S}(\mathbb{R})\to \mathcal{S}(\mathbb{R}),f\mapsto f\circ \phi$, in the case that φ is a non constant polynomial. Recall that $\mathcal{S}(\mathbb{R})$ consists of those smooth functions $f:\mathbb{R}\to \mathbb{C}$ with the property that

$${\pi }_n(f):=\underset{x\in \mathbb{R}}{\sup }\underset{0\le j\le n}{\sup }{(1+x^2)}^n|f^{(j)}(x)|<\infty$$

for each $n\in \mathbb{N}$. $\mathcal{S}(\mathbb{R})$ is a Fréchet Montel space when endowed with the topology generated by the sequence of seminorms ${\left({\pi }_n\right)}_{n\in \mathbb{N}}$. In particular, a sequence ${(f_j)}_j$ in $\mathcal{S}(\mathbb{R})$ converges to f if and only if $P(x){(f_j-f)}^{(n)}(x)$ converges to zero uniformly on the real line for each polynomial P and each $n\in \mathbb{N}$. This space, introduced by Schwartz [20], plays a central role in the theory of Linear Partial Differential Equations and also in Harmonic Analysis (see for example [11]). We refer to [14] for background on Fréchet spaces.

A smooth function $\phi :\mathbb{R}\to \mathbb{R}$ is said to be a symbol for $\mathcal{S}(\mathbb{R})$ if $C_{\phi }$ maps $\mathcal{S}(\mathbb{R})$ continuously into itself. The symbols for $\mathcal{S}(\mathbb{R})$ were completely characterized in [8, Theorem 2.3]. It follows from that characterization that any non constant polynomial φ is a symbol for $\mathcal{S}(\mathbb{R})$. The results of the present paper complement our study in [7] where we investigate dynamics and the spectrum of some particular composition operators. Concrete examples were given where the spectrum coincides with the open unit disc, the unit circle or $\mathbb{C}\setminus \{0\}$. In particular, the spectrum of translation and dilation operators was analyzed in [7, Examples 5-6]:

Example 1.1

• Let $\phi (x)=x+1$. Then $\sigma \left(C_{\phi }\right)=\{\lambda \in \mathbb{C}:|\lambda |=1\}$.

• Let $\phi (x)=ax$ where $a\ne 0$ and $|a|\ne 1$. Then $\sigma \left(C_{\phi }\right)=\mathbb{C}\setminus \{0\}$.

However, in [7] we did not obtain any result concerning the spectrum of the composition operator with a polynomial of degree greater than one. This is precisely the objective of this work. It turns out that some dynamical properties of the composition operator are characterized by the spectrum of the operator. For example, the spectrum of a mean ergodic composition operator is always contained in the closed unit disk ([7, Corollary 4.5]). This result can be improved when the symbol is a polynomial. As we prove in Theorem 2.5, the spectrum of a composition operator which is mean ergodic and whose symbol is a polynomial with degree greater than one coincides with {0} while the behavior of non mean ergodic composition operators with polynomial symbols is different (Theorem 2.8). For strictly decreasing symbols, not necessarily polynomials, the containment of $\partial \mathbb{D}$ in the spectrum of the operator is equivalent to its mean ergodicity.

In [5] the spectrum of composition operators on $\mathcal{A}(\mathbb{R})$, the space of all real analytic functions, is investigated for the case that the symbol is a quadratic polynomial. For quadratic polynomials, we have a complete characterization of the spectrum of the corresponding composition operator depending on the number of fixed points of the polynomial. As expected, the spectrum of the operator depends on the space where it is considered. To give an example, when φ is a quadratic polynomial without fixed points then ${\sigma }_{\mathcal{A}(\mathbb{R})}(C_{\phi })=\mathbb{C}$, whereas ${\sigma }_{\mathcal{S}(\mathbb{R})}(C_{\phi })=\{0\}$.

Theorem 4.1 contains a complete description of the spectrum of a composition operator with a polynomial of degree three whose leading coefficient is positive. For polynomials with negative leading coefficient some partial results are available but we lack a complete characterization.

The final section contains some results concerning the spectra of composition operators with (non polynomial) monotone symbols.

We recall that given an operator $T:X\to X$ on a Fréchet space X, $\sigma (T)$, the spectrum of T, is the set of all $\lambda \in \mathbb{C}$ such that $T-\lambda I:X\to X$ does not admit a continuous linear inverse. Unlike for Banach spaces, the spectrum $\sigma (T)$ need not be compact and it may happen that $\sigma (T)=\varnothing$. An operator T is said to be power bounded if $\{T^n(x):n\in \mathbb{N}\}$ is bounded for each $x\in X$. A closely related concept to power boundedness is that of mean ergodicity. Given $T\in L(X)$, the Cesàro means of T are defined as $T_{[n]}={\sum }_{k=1}^n{T}^k/n$. T is said to be mean ergodic when $T_{[n]}$ converges to an operator P, which is always a projection, in the strong operator topology, i.e. if $(T_{[n]}(x))$ is convergent to $P(x)$ for each $x\in X$. There is a vast literature studying dynamics of linear operators on Banach or Fréchet spaces. See for instance [1], [4], [10], [13], [15] and the references therein.

From now on ${\phi }_n=\phi \circ \dots \circ \phi$ denotes the n-th iteration of φ.

The following results will be used in what follows.

Lemma 1.2

[7, 3.10] Let φ be a polynomial of even degree without fixed points. Then there is $N\in \mathbb{N}$ such that $\psi ={\phi }_N$ has neither zeros nor fixed points. Moreover, for every $K>0$ there is $m_0\in \mathbb{N}$ such that

$$|{\psi }_{m+1}(t)|\ge K{({\psi }_m(t))}^2\forall m\ge m_0,\forall t\in \mathbb{R}.$$

Theorem 1.3

[7, 3.11] Let φ be a polynomial with degree greater than or equal to two. Then, the following are equivalent:

• $C_{\phi }$ is power bounded.

• $C_{\phi }$ is mean ergodic.

• The degree of φ is even and it has no fixed points.