Iterates of composition operators on global spaces of ultradifferentiable functions

Abstract We analyze the behaviour of the iterates of composition operators defined by polynomials acting on global classes of ultradifferentiable functions of Beurling type and being invariant under Fourier transform. We characterize the polynomials $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> for which the sequence of iterates is equicontinuous between two different Gelfand–Shilov spaces. For the particular case in which the weight $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> is equivalent to a power of the logarithm, the result obtained characterizes the polynomials $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> for which the composition operator $$C_\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> is power bounded in $${{\mathcal {S}}}_\omega ({{\mathbb {R}}}).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>ω</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Unlike the composition operators in the Schwartz class, the Waelbroek spectrum of an operator $$C_\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> , $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> being a polynomial of degree greater than one lacking fixed points is never compact. We focus on the problem of the convergence of Neumann series. We deduce the continuity of the resolvent operator between two different Gelfand–Shilov classes for polynomials $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> lacking fixed points. Concerning polynomials of second degree the most interesting case is the one in which the polynomial only has one fixed point: we provide some restrictions on the indices $$d, d'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> that are necessary for the resolvent operator to be continuous between the Gelfand–Shilov classes $$\Sigma _d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> and $$\Sigma _{d'}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Σ</mml:mi> <mml:msup> <mml:mi>d</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>