Singly generated invariant subspaces in the Hardy space on the unit ball

In this paper, we give a complete characterization of singly generated invariant subspaces in the Hardy spaceon the unit ball. Then we construct a singly generated invariant subspace that cannot be generated by a single innerfunction, contrary to the one-variable case where every invariant subspace is generated by a single inner function. Someimportant properties of invariant subspaces are also determined for singly generated invariant subspaces.

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[1] Agrawal OP, Clark DN, Douglas RG. Invariant subspaces in the polydisk. Pacific Journal of Mathematics 1986; 121: 1-11.
[2] Aleksandrov AB. Existence of inner functions in the unit ball. Matemticheskii Sbornik 1982; 118 (160): 147-163.
[3] Beurling A. On two problems concerning linear transformations in Hilbert space. Acta Mathematica 1949; 81: 239-255.
[4] Chen XM, Guo KY. Analytic Hilbert Modules. Boca Raton, FL, USA: Chapman Hall/CRC, 2003.
[5] Douglas R, Paulsen V. Hilbert Modules over Function Algebras. Pitman Research Notes in Mathematics Series. Boston, MA, USA: Longman Scientific & Technical, 1989.
[6] Gowda M. Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of Cn . Transactions of the American Mathematical Society 1983; 277 (1): 203-212.
[7] Koca BB, Sadık N. Invariant subspaces generated by a single function in the polydisc. Math Notes 2017; 102: 193-197.
[8] Løw E. A construction of inner functions on the unit ball in Cp . Inventiones Mathematicae 1982; 67: 223-229.
[9] Rudin W. Function Theory in Polydiscs. New York, NY, USA: W. A. Benjamin, Inc., 1969.
[10] Rudin W. Zeros of holomorphic functions in balls. Indagationes Mathematicae 1976; 79 (1): 57-65.
[11] Rudin W. Function Theory in the Unit Ball of Cn . New York, NY, USA: Springer-Verlag, 1980.
[12] Rudin W. Inner functions in the unit ball of Cn . Journal of Functional Analysis 1983; 50 (1): 100-126.
[13] Zhu K. Spaces of Holomorphic Functions in the Unit Ball. New York, NY, USA: Springer-Verlag, 2005.