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# Journal of Operator Theory

Volume 61, Issue 1, Winter 2009  pp. 75-86.

Infinite sequences of inner functions and submodules in $H^2({\mathbb D}^2)$

Authors Michio Seto
Author institution: Department of Mathematics, Shimane University, 1060 Nishi-Kawatsu, Matsue, Shimane 690-8504, Japan

Summary:  We deal with infinite sequences of inner functions $\{q_j\}_{j\geqslant 0}$ with the property that {\it $q_{j}$ is divisible by $q_{j+1}$}. It is shown that these sequences have a close relation to the module structure of the Hardy space over the bidisk. Commutators, Hilbert--Schmidt norms and spectra of operators related to the module structure will be calculated exactly.

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