A special subspace of weighted spaces of holomorphic functions on the upper half plane
Subject Areas : Statistics
1 - Dept. of mathematics, Faculty of Science, University of Kurdistan, Sanandaj , Iran
Keywords: فضای باناخ, قضیه رستهایی بئر, نرم سوپریمم وزنی, مجموعه F_σ,
Abstract :
In this paper, we intend to define and study concepts of weight and weighted spaces of holomorphic (analytic) functions on the upper half plane. We study two special classes of these spaces of holomorphic functions on the upper half plane. Firstly, we prove these spaces of holomorphic functions on the upper half plane endowed with weighted norm supremum are Banach spaces. Then, we investigate these spaces of holomorphic functions on the upper half plane from a new aspect which has not been considered up to now. Indeed we prove that without any necessary condition on a weight such as restricting the rate of growth from below or above (constructing the upper bound or lower bound) or limit condition (except the continuity on the upper half plane) any weighted spaces of holomorphic functions on the upper half plane has a special subspace which can be written as countable intersection of closed sets.
[1] A.L.Shields, D.L. Williams, Bounded projections and the growth of harmonic conjugates in the disc, Michigan Math. J. 3(1982), 3-25.
[2] A.L.Shields, D.L. Williams, Bounded projections, duality, and multipliers in spaces of harmonic functions, J. Reine. Angew Math. 299/300 (1978), 265-279.
[3] A.L.Shields, D.L. Williams, Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287-302
[4] W.Lusky, on the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia Math. 1(2006), 19-45.
[5] W.Lusky, on weighted spaces of harmonic and holomorphic functions, J. Lond. Math Soc. 51(1995), 309-320.
[6] W.Lusky, Growth conditions for harmonic and holomorphic functions, Functional Analysis (Trier, 1994), S. Dierolfetal. (End), de Gruyter (1996), 281-291.
[7] J.Bonet, P.Domanski, M.Lindstrom, pointwise multiplication operators on weighted Banach spaces of analytic functions, Studia math. 137(1999), 177-194.
[8] J.Bonet, P.Domanski, M.Lindstrom, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. Math. Bull. 42(1999), 139-148.
[9] M.A.Ardalani and W.Lusky, Bounded operators on weighted spaces of holomorphic functions on upper half plane, Studia math. 209(2012), 225-234.
[10] M.A.Ardalani and W.Lusky, weighted spaces of holomorphic functions on the upper half-plane math, Scandinavica, 111(2012), 224-260.
[11] S. Lang, Complex Analysis, Springer Verlag, New York, 1999.