In the first part of this study, we characterize the compact subspaces of Hpu(Bp) and their relation to the vanishing Carleson measures. In the second part, we discuss the dual complement of the complex ellipsoid and give a duality result for Hpu(Bp)spaces in the sense of Grothendieck–Köthe–da Silva. | Turk J Math (2018) 42: 2157 – 2165 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Compactness and Duality on Poletsky–Stessin Hardy Spaces of Complex Ellipsoids Sibel ŞAHİN∗, Department of Mathematics, Mimar Sinan Fine Arts University Received: • Accepted/Published Online: • Final Version: Abstract: In the first part of this study, we characterize the compact subspaces of Hup (Bp ) and their relation to the vanishing Carleson measures. In the second part, we discuss the dual complement of the complex ellipsoid and give a duality result for Hup (Bp ) spaces in the sense of Grothendieck–Köthe–da Silva. Key words: Duality, dual complement, Poletsky–Stessin Hardy space, compactness, complex ellipsoid 1. Introduction In their seminal work [4], Poletsky and Stessin showed that it is possible to generalize the whole idea of Hardy and Bergman spaces in the general context of hyperconvex domains in higher dimensions. After this leading work, in [5, 6] we concentrated on these generalized spaces in various domains but especially complex ellipsoids and in the present work of continuation, we consider the compactness and Grothendieck–Köthe–da Silva duality properties of these spaces. Firstly, we try to identify the characteristics of compact subspaces of these generalized Hardy spaces, and then relate these properties with vanishing Carleson measures and compact linear operators. Secondly, we describe the dual complements of complex ellipsoids and give a duality result analogous to [2] in the sense of Grothendieck–Köthe–da Silva. The organization of the present paper is as follows: In Section 2, we recall the Poletsky–Stessin Hardy spaces, Hup (Bp ) , on the complex ellipsoid Bp and we introduce the Cauchy–Fantappie integral associated with the Monge–Ampère measure µu , together with an integral representation for Hup (Bp ). The main results of the present study .
