Nihonkai Mathematical Journal

One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk II

Kei Ji Izuchi, Kou Hei Izuchi, and Yuko Izuchi

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Abstract

This paper is a continuation of the previous paper [9]. Let $M_1$ be an invariant subspace of $H^2$ over the bidisk. Then there exists a nonzero $f_0$ in $M_1$ such that $M_2:=M_1\ominus \mathbb{C} \cdot f_0$ is also an invariant subspace. A relationship is given the ranks of the cross commutators $[R^*_z,R_w]$ on $M_1$ and $M_2$. We also give a relationship of the ranks of the cross commutators $[S_w,S^*_z]$ on $H^2\ominus M_1$ and $H^2\ominus M_2$.

Note

The first author is supported by JSPS KAKENHI Grant Number 15K04895.

Article information

Source
Nihonkai Math. J., Volume 28, Number 1 (2017), 31-42.

Dates
Received: 24 December 2015
Revised: 11 May 2016
First available in Project Euclid: 7 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1520391678

Mathematical Reviews number (MathSciNet)
MR3771366

Zentralblatt MATH identifier
06714331

Subjects
Primary: 47A15: Invariant subspaces [See also 47A46] 32A35: Hp-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15]
Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

Keywords
Hardy space over the bidisk invariant subspace one dimensional perturbation rank of operator cross commutator

Citation

Izuchi, Kei Ji; Izuchi, Kou Hei; Izuchi, Yuko. One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk II. Nihonkai Math. J. 28 (2017), no. 1, 31--42. https://projecteuclid.org/euclid.nihmj/1520391678


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References

  • P. Ghatage and V. Mandrekar, On Beurling type invariant subspaces of $L^2(T^2)$ and their equivalence, J. Operator Theory 20 (1988), 83–89.
  • K. J. Izuchi and K. H. Izuchi, Rank-one commutators on invariant subspaces of the Hardy space on the bidisk, J. Math. Anal. Appl. 316 (2006), 1–8.
  • K. J. Izuchi and K. H. Izuchi, Cross commutators on backward shift invariant subspaces over the bidisk, Acta Sci. Math. (Szeged) 72 (2006), 251–270.
  • K. J. Izuchi and K. H. Izuchi, Rank-one commutators on invariant subspaces of the Hardy space on the bidisk II, J. Operator Theory 60 (2008), 239–251.
  • K. J. Izuchi and K. H. Izuchi, Rank-one cross commutators on backward shift invariant subspaces on the bidisk, Acta Math. Sin. (Engl. Ser.) 25 (2009), 693–714.
  • K. J. Izuchi and K. H. Izuchi, Ranks of cross commutators on backward shift invariant subspaces over the bidisk, Rocky Mountain J. Math. 40 (2010), 929–942.
  • K. J. Izuchi and K. H. Izuchi, Cross commutators on backward shift invariant subspaces over the bidisk II, J. Korean Math. Soc. 49 (2012), 139–151.
  • K. J. Izuchi and K. H. Izuchi, Commutativity in two-variable Jordan blocks on the Hardy space, Acta Sci. Math. (Szeged) 78 (2012), 129–136.
  • K. J. Izuchi, K. H. Izuchi, and Y. Izuchi, One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk I, Nihonkai Math. J. 28 (2017), 1–29.
  • K. J. Izuchi, T. Nakazi, and M. Seto, Backward shift invariant subspaces in the bidisc II, J. Operator Theory 51 (2004), 361–376.
  • K. J. Izuchi, T. Nakazi, and M. Seto, Backward shift invariant subspaces in the bidisc III, Acta Sci. Math. (Szeged) 70 (2004), 727–749.
  • V. Mandrekar, The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc. 103 (1988), 145–148.
  • T. Nakazi, Invariant subspaces in the bidisc and commutators, J. Austral. Math. Soc. (Ser. A) 56 (1994), 232–242.