A note on classification of submodules in 𝐻²(𝐷²)

Yang, Rongwei

doi:10.1090/S0002-9939-09-09893-1

A note on classification of submodules in $H^{2}(D^{2})$
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Proc. Amer. Math. Soc. 137 (2009), 2655-2659 Request permission

Abstract:

The Hardy spaces $H^{2}(D^{2})$ can be viewed as a module over the polynomial ring $C[z_1,z_2]$. Based on a study of core operator, a new equivalence relation for submodules, namely congruence, was introduced. This paper gives a classification of congruent submodules by the rank of core operators.

References

Xiaoman Chen and Kunyu Guo, Analytic Hilbert modules, Chapman & Hall/CRC Research Notes in Mathematics, vol. 433, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1988884, DOI 10.1201/9780203008836
R. G. Douglas, V. I. Paulsen, C.-H. Sah, and K. Yan, Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math. 117 (1995), no. 1, 75–92. MR 1314458, DOI 10.2307/2375036
Kunyu Guo and Rongwei Yang, The core function of submodules over the bidisk, Indiana Univ. Math. J. 53 (2004), no. 1, 205–222. MR 2048190, DOI 10.1512/iumj.2004.53.2327
Keiji Izuchi and Kou Hei Izuchi, Rank-one commutators on invariant subspaces of the Hardy space on the bidisk, J. Math. Anal. Appl. 316 (2006), no. 1, 1–8. MR 2201744, DOI 10.1016/j.jmaa.2005.04.021
Keiji Izuchi, Takahiko Nakazi, and Michio Seto, Backward shift invariant subspaces in the bidisc. II, J. Operator Theory 51 (2004), no. 2, 361–376. MR 2074186
Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
Rongwei Yang, On two-variable Jordan blocks, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 739–754. MR 2034205
Rongwei Yang, The core operator and congruent submodules, J. Funct. Anal. 228 (2005), no. 2, 469–489. MR 2175415, DOI 10.1016/j.jfa.2005.06.022