Abstract
The invariant subspaces of the Hardy space \(H^2(\mathbb {D})\) of the unit disc are very well known; however, in several variables, the structure of the invariant subspaces of the classical Hardy spaces is not yet fully understood. In this study, we examine the structure of invariant subspaces of Poletsky–Stessin–Hardy spaces which are the generalization of the classical Hardy spaces to hyperconvex domains in \(\mathbb {C}^n\). We showed that not all invariant subspaces of \(H^{2}_{\tilde{u}}(\mathbb {D}^2)\) are of Beurling-type. To characterize the Beurling-type invariant subspaces of this space, we first generalized the Lax–Halmos Theorem to the vector-valued Poletsky–Stessin–Hardy spaces and then we gave a necessary and sufficient condition for the invariant subspaces of \(H^{2}_{\tilde{u}}(\mathbb {D}^2)\) to be of Beurling-type.
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Alan, M.A., Göǧüş, N.G.: Poletsky–Stessin Hardy spaces in the plane. Complex Anal. Oper. Theory 8(5), 975–990 (2014)
Aytuna, A.: Some results on HP-Spaces on strictly Pseudoconvex Domains. PhD Dissertation, University of Washington (1976)
Beurling, A.: On two problems concerning linear transformations in Hilbert space. Acta Math. 81, 17 (1948)
Conway, J.B.: A course in functional analysis, 2nd edn. Springer-Verlag, New York (1990)
Demailly, J.P.: Mesures de Monge–Ampère et Caractérisation Géométrique des Variétés Algébraiques Affines. Mémoire de la Société Mathématique de France 19, 1–124 (1985)
Duren, P.L.: Theory of HP spaces. Academic Press Inc., New York, London (1970)
Halmos, P.R.: A Hilbert Space Problem Book. Graduate texts in mathematics, 2nd edn. Springer-Verlag, New York, Berlin (1982)
Jacewicz, C.A.: A nonprincipal invariant subspace of the Hardy space on the torus. Proc. Am. Math. Soc. 31, 127–129 (1972)
Poletsky, E.A., Stessin, M.I.: Hardy and Bergman spaces on Hyperconvex domains and their composition operators. Indiana Univ. Math. J. 57, 2153–2201 (2008)
Radjavi, H., Rosenthal, P.: Invariant Subspaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 77. Springer-Verlag, New York, Heidelberg (1973)
Rosenblum, M., Rovnyak, J.: Hardy classes and operator theory. Oxford Mathematical Monographs. Oxford Science PublicationsOxford Science PublicationsOxford Science PublicationsOxford Science Publications. The Clarendon Press, Oxford University Press, New York (1985)
Rudin, W.: Function Theory in Polydiscs, p. vii+188. W. A. Benjamin Inc, New York, Amsterdam (1969)
Sadikov, N.M.: Invariant subspaces in the Hardy space on a bidisk. Spectr. Theory Oper. Appl. 7, 186–200 (1986)
Şahin, S.: Monge–Ampère measures and Poletsky–Stessin Hardy spaces on bounded hyperconvex domains. PhD Dissertation, Sabancı University, (2014)
Şahin, S.: Poletsky–Stessin Hardy spaces on domains bounded by an snalytic Jordan curve in \(\mathbb{C}\). Compl. Var. Elliptic Equ. 60(8), 1114–1132 (2015)
Shresta, K.: Poletsky–Stessin Hardy spaces on the unit disk. PhD Dissertation, Syracuse University, Dissertations-ALL. Paper 279 (2015)
Sz.-Nagy, B., Foias, C.: Harmonic Analysis of Operators on Hilbert Space. Akademiai Kiadó Budapest (1970)
Yang, R.: A Brief Survey of Operator Theory in \(H^2(\mathbb{D}^2)\), Handbook of analytic operator theory, 223–258. Handb. Math. Ser, CRC Press/Chapman Hall, Boca Raton (2019)
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Communicated by Anton Baranov.
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Eskişehirli, B.B., Şahin, S. Beurling-type invariant subspaces of the Poletsky–Stessin–Hardy spaces in the bidisc. Ann. Funct. Anal. 12, 43 (2021). https://doi.org/10.1007/s43034-021-00131-y
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DOI: https://doi.org/10.1007/s43034-021-00131-y