Characterization of composition operators with closed range for one-dimensional smooth symbols

Composition operators on the Schwartz space

“…Our aim is to obtain necessary or sufficient conditions for the range of the composition operator to be closed in S(R). We will relate the closed range property of a composition operator C ϕ : S(R) → S(R) with the closed range property of C ϕ : C ∞ (R) → C ∞ (R) (characterized by [10]) and the closed range property of multiplication operators on S(R), which has been characterized in [1]. Concrete examples of composition operators on S(R) lacking the closed range property are provided.…”

“…In particular, we prove that C ϕ can never be a compact operator and obtain necessary or sufficient conditions for the range of the composition operator to be closed in S(R). These conditions are expressed in terms of the multiplication operator studied in [1] and the closed range of the corresponding operator considered in the space of smooth functions, involving then the conditions considered by Przestacki in [9,10,11]. We remark that the characterization of the symbols is valid for the several variables case.…”

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

“…Since r > 1 is arbitrary we conclude that for every r > 1 there exist C > 0 and q ∈ N such that |ϕ ′′ m (x)| ≤ Cr m (1 + ϕ m (x)) q whenever x ≥ 1 2 . For n > 2 we apply (10) to get…”

“…for each x ∈ R and each m ∈ N. Since ϕ is an odd function, it suffices to consider x ≥ 0. First, we check the inequality (11) for the first derivative.…”

“…Composition operators on Fréchet spaces of smooth functions on the reals have attracted the attention of several authors recently ( [7,5,8,9,10,11]) but to our knowledge very little is known about the spectra of composition operators in this setting. See for instance [3] or [2], where the spectrum of composition operators and other classical operators on spaces of real smooth functions is investigated.…”