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Nψ,ϕ-type Quotient Modules over the Bidisk

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Abstract

Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.

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References

  1. Arveson, W.: The Dirac operator of a commuting d-tuple. J. Funct. Anal., 189, 53–79 (2002)

    Article  MathSciNet  Google Scholar 

  2. Brown, L., Douglas, R., Fillmore, P.: Extension of C*-algebras and K-homology. Ann. of Math., 105, 265–324 (1977)

    Article  MathSciNet  Google Scholar 

  3. Clark, D.: Restrictions of Hp functions in the polydisk. Amer. J. Math., 110, 1119–1152 (1988)

    Article  MathSciNet  Google Scholar 

  4. Douglas, R.: Essentially reductive Hilbert modules. J. Operator Theory, 55, 117–133 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Douglas, R., Misra, G.: Some calculations for Hilbert modules. J. Orissa Math. Soc., 12–15, 75–85 (1993–1996)

    Google Scholar 

  6. Izuchi, K. J., Izuchi, K. H., Izuchi, Y.: Wandering subspaces and the Beurling type theorem. II. New York J. Math., 16, 489–505 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Izuchi, K., Yang, R.: Nϕ-type quotient modules on the torus. New York J. Math., 14, 431–457 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Wang, P.: The essential normality of type quotient module of Hardy module on the polydisc. Proc. Amer. Math. Soc., 142, 151–156 (2014)

    Article  MathSciNet  Google Scholar 

  9. Wang, P., Zhao, C.: Essential normality of homogenous quotient modules over the polydisc: distinguished variety case. Integr. Equ. Oper. Theory, 90, 13 (2018)

    Article  MathSciNet  Google Scholar 

  10. Wang, P., Zhao, C.: Essentially normal homogeneous quotient modules on the polydisc. Adv. Math., 339, 404–425 (2018)

    Article  MathSciNet  Google Scholar 

  11. Wu, C., Yu, T.: Wandering subspace property of the shift operator on a class of invariant subspaces of the weighted Bergman space L 2a (dA2). Banach J. of Math. Anal., https://doi.org/10.1007/s43037-019-00039-9

  12. Yang, R.: Operator theory in the Hardy space over the bidisk. II. Integr. Equ. Oper. Theory, 42, 99–124 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to professor Rongwei Yang for suggesting this line of research, and for his ongoing encouragement and support.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P. R. China

    Chang Hui Wu

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P. R. China

    Tao Yu

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  1. Chang Hui Wu
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  2. Tao Yu
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Correspondence to Tao Yu.

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Supported by NNSF of China (Grant No. 11971087)

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Wu, C.H., Yu, T. Nψ,ϕ-type Quotient Modules over the Bidisk. Acta. Math. Sin.-English Ser. 36, 943–960 (2020). https://doi.org/10.1007/s10114-020-9347-8

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  • DOI: https://doi.org/10.1007/s10114-020-9347-8

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