Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CROSS COMMUTATORS ON BACKWARD SHIFT INVARIANT SUBSPACES OVER THE BIDISK II

  • Received : 2010.09.20
  • Published : 2012.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

In the previous paper, we gave a characterization of backward shift invariant subspaces of the Hardy space over the bidisk on which [${S_z}^n$, $S_w^*$] = 0 for a positive integer n ${\geq}$ 2. In this case, it holds that ${S_z}^n=cI$ for some $c{\in}\mathbb{C}$. In this paper, it is proved that if [$S_{\varphi}$, $S_w^*$] = 0 and ${\varphi}{\in}H^{\infty}({\Gamma}_z)$, then $S_{\varphi}=cI$ for some $c{\in}\mathbb{C}$.

Keywords

References

  1. H. Bercovici, Operator Theory and Arithmetic in $H^{\infty}$, Mathematical Surveys and Monographs, 26. American Mathematical Society, Providence, RI, 1988.
  2. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239-255.
  3. X. Chen and K. Guo, Analytic Hilbert Modules, Chapman & Hall/CRC, Boca Raton, FL, 2003.
  4. K. J. Izuchi and K. H. Izuchi, Commutativity in two variable Jordan blocks on the Hardy space, preprint.
  5. K. J. Izuchi and K. H. Izuchi, Cross commutators on backward shift invariant subspaces over the bidisk, Acta Sci. Math. (Szeged) 72 (2006), no. 1-2, 251-270.
  6. K. J. Izuchi, T. Nakazi, and M. Seto, Backward shift invariant subspaces in the bidisc. II, J. Operator Theory 51 (2004), no. 2, 361-376.
  7. K. J. Izuchi, T. Nakazi, and M. Seto, Backward shift invariant subspaces in the bidisc. III, Acta Sci. Math. (Szeged) 70 (2004), no. 3-4, 727-749.
  8. V. Mandrekar, The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc. 103 (1988), no. 1, 145-148. https://doi.org/10.1090/S0002-9939-1988-0938659-7
  9. N. Nikol'skii, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.
  10. W. Rudin, Function Theory in Polydiscs, Benjamin, New York, 1969.
  11. R. Schneider, Isometries of $H^p(U^n)$, Canad. J. Math. 25 (1973), 92-95. https://doi.org/10.4153/CJM-1973-007-0