Abstract
This paper gives some estimates of the essential norm for the difference of composition operators induced byφandψacting on the space,H∞(Dn), of bounded analytic functions on the unit polydiscDn, whereφandψare holomorphic self-maps ofDn. As a consequence, one obtains conditions in terms of the Carathéodory distance onDnthat characterizes those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators onH∞(Dn)is compact.
Highlights
Let Dn be the unit polydisc of Cn with boundary ∂Dn
We give some upper and lower estimates of the essential norm for the difference of composition operators induced by φ and ψ acting on the space H∞ Dn, where φ and ψ are analytic self-maps of Dn
One obtains conditions in terms of the Cartheodory distance on Dn that characterize those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators on H∞ Dn is compact
Summary
Let Dn be the unit polydisc of Cn with boundary ∂Dn. If n 1, we will denote the unit disk D1 by D. In the past few decades, boundedness, compactness, and essential norms of composition and closely related operators between various spaces of holomorphic functions have been studied by many authors see, e.g., the following papers mostly in the settings of the unit ball and the unit polydisc 1–23 and the references therein. Differences of composition operators on the Bloch and the little Bloch space are studied in 30, 31 Motivated by these results, we give some upper and lower estimates of the essential norm for the difference of composition operators induced by φ and ψ acting on the space H∞ Dn , where φ and ψ are analytic self-maps of Dn. As a consequence, one obtains conditions in terms of the Cartheodory distance on Dn that characterize those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators on H∞ Dn is compact
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