Spectra of Composition Groups on the Weighted Dirichlet Space of the Upper Half-Plane
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
SPECTRA OF COMPOSITION GROUPS ON THE WEIGHTED DIRICHLET SPACE OF THE UPPER HALF-PLANE∗ M.O. AGWANG
J.O. BONYO†
Department of Pure and Applied Mathematics, Maseno University, P.O. BOX 333-40105, Maseno, Kenya E-mail : [email protected]; [email protected] Abstract We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane U is strongly continuous on the weighted Dirichlet space of U, Dα (U). Further, we investigate when they are isometries on Dα (U). In each case, we determine the semigroup properties while in the case that the induced composition group is an isometry, we apply similarity theory to determine the spectral properties of the group. Key words
one-parameter semigroup; composition operator semigroups; strong continuity; infinitesimal generator; spectra
2010 MR Subject Classification
1
47B38; 47D03; 47A10
Introduction
For an open subset Ω of C, let H(Ω) denotes the Fr´echet space of analytic functions on Ω endowed with the topology of uniform convergence on compact subsets of Ω. In this note, Ω can be either the open unit disk D or the upper half plane U. If ϕ is a self analytic map on Ω, then the induced composition operator Cϕ acting on H(Ω) is defined by Cϕ f = f ◦ ϕ, with the corresponding weighted composition operator on H(Ω) given by Sϕ = (ϕ′ )γ Cϕ for some appropriately chosen weight γ. Composition operators on spaces of analytic functions on the unit disc H(D) have been extensively studied in the literature comparatively to their counterparts on the analytic spaces of the upper half plane H(U). Even though there are isomorphisms between the corresponding spaces of D and of U, composition operators act differently in the two cases. For instance, unlike the case of Hardy or Bergman spaces of D, not every composition operator is bounded on Hardy or Bergman spaces of U, see [1, 2]. It has also been proved in [3, 4] that there are no non-trivial (i.e. with symbol not constant) compact composition operators on the Hardy space H 2 (U) or the weighted Bergman space L2a (U, µα ) which is not the case for H 2 (D) or L2a (D, mα ). ∗ Received
July 18, 2019; revised March 14, 2020. This work was partially supported by a grant from the Simons Foundation. † Corresponding author: J.O. BONYO.
1740
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
Unlike the Hardy and Bergman spaces of D cases, composition operators on the (weighted) Dirichlet space of the unit disk Dα (D) are not necessarily bounded; an indication that the action of composition operators on the weighted Dirichlet space of the upper half plane Dα (U) is very much complicated. Recent attempts to study composition operators on Dα (U) can be found in [5, 6]. Schroderus [5] obtained the spectrum of composition operators induced by linear fractional transformations (LFTs) of the upper half plane U; while Sharma, Sharma and Raj [6] characterized boundedness and compactness of comp
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