tailieunhanh - Báo cáo hóa học: " A note on the Königs domain of compact composition operators on the Bloch space"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A note on the Königs domain of compact composition operators on the Bloch space | Jones Journal of Inequalities and Applications 2011 2011 31 http content 2011 1 31 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access A note on the Konigs domain of compact composition operators on the Bloch space Matthew M Jones 1 Correspondence . Department of Mathematics Middlesex University The Burroughs London NW4 4BT UK SpringerOpen0 Abstract Let D be the unit disk in the complex plane. We define B0 to be the little Bloch space of functions f analytic in D which satisfy lim z 1 1 - z 2 f z 0. If p D Dis analytic then the composition operator Cộ f w f Ộ is a continuous operator that maps B0 into itself. In this paper we show that the compactness of Cộ as an operator on B0 can be modelled geometrically by its principal eigenfunction. In particular under certain necessary conditions we relate the compactness of Cộ to the geometry of ơ D where Ơ satisfies Schoder s functional equation ơ Ộ 0 ấ 2000 Mathematics Subject Classification Primary 30D05 47B33 Secondary 30D45. 1 Introduction Let D z e C z 1 be the unit disk in the complex plane and T its boundary. We define the Bloch space B to be the Banach space of functions f analytic in D with f b If 0 sup 1 - z 2 f z ra. zeD This space has many important applications in complex function theory see 1 for an overview of many of them. We denote by B0 the little Bloch space of functions in B that satisfy lim z 1 1 - z 2 f z 0. This space coincides with the closure of the polynomials in B. Suppose now that p D D is analytic then we may define the operator Cộ acting on B0 as f w f Ộ. It was shown in 2 that every such operator maps B0 continuously into itself. Moreover it was proved that Cộ is compact on B0 if and only if Ộ satisfies 1 - z 2 lim ----- pfz 0. 1 z 1 1 - p z 2 Recall that the hyperbolic geometry on D is defined by the distance disk z w inf ị ẤD d where the infimum is taken over all sufficiently smooth arcs that .