Composition operators on function spaces on the halfplane: spectra and semigroups (Q6610394)

The paper involves properties of composition operators on holomorphic function spaces on the right half-plane \(\mathbb C_+=\{z=x+iy: x>0\}\), both as individual operators or as elements of one-parameter semigroups. The authors analyze the so-called Zen spaces (weighted Hardy-Bergman spaces) \(A^2_\nu\) defined to consist of all analytic functions \(F\) on \(\mathbb C_+\) such that \(\|F\|^2=\sup_{\varepsilon >0}\int_{\overline{\mathbb C_+}}|F(z+\varepsilon)|^2 d\nu(x)dy<\infty\), where \(\nu\) is a positive regular Borel measure on \([0,\infty)\) satisfying the doubling condition \(\sup_{t>0}\frac{\nu([0,2t)}{\nu[0,t)}<\infty\). Covering the cases \(H^2(\mathbb C_+)\) for \(\nu=\delta_0\) and \(A^2(\mathbb C_+)\) for \(d\nu(x)=dx\). They complete many results known in the particular cases of Hardy and Bergman spaces on the right-half plane. In particular estimating the norm and essential norm of composition operators \(C_\phi\), generalizing the results by \textit{A. S. Kucik} [Complex Anal. Oper. Theory 12, No. 8, 1817--1833 (2018; Zbl 06984208)], results on the spectrum and the essential spectrum of the associated composition operator \(C_\phi\) extending those proved [\textit{R. Schroderus}, J. Math. Anal. Appl. 447, No. 2, 817--833 (2017; Zbl 1358.47016)] and also generalizing to Zen spaces the result of \textit{A. G. Arvanitidis} [Acta Sci. Math. 81, No. 1--2, 293--308 (2015; Zbl 1363.47078); Acta Sci. Math. 89, No. 3--4, 635 (2023; Zbl 07794415)] concerning description of boundedness of a semigroup \((C_{\phi_t})\). Their results are illustrated by considering a new case, the intersection of the Hardy and Bergman Hilbert spaces on the half-disc.\N\NFor the entire collection see [Zbl 1531.47001].