Nihonkai Mathematical Journal

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

Kei Ji Izuchi, Kou Hei Izuchi, and Yuko Izuchi

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Abstract

For an invariant subspace $M_1$ of the Hardy space $H^2$ over the bidisk $\mathbb{D}^2$, write $N_1=H^2 \ominus M_1$. Let $\Omega(M_1)=M_1\ominus(z M_1+w M_1)$ and $\widetilde\Omega(N_1)=\{f\in N_1: z f, w f\in M_1\}$. Then $\Omega(M_1)\not=\{0\}$, and $\Omega(M_1), \widetilde\Omega(N_1)$ are key spaces to study the structure of $M_1$. It is known that there is a nonzero $f_0\in M_1$ such that $M_2=M_1\ominus \mathbb{C} \cdot f_0$ is an invariant subspace. It is described the structures of $\Omega(M_2), \widetilde\Omega(N_2)$ using the words of $\Omega(M_1), \widetilde\Omega(N_1)$ and $f_0$. To do so, it occur many cases. We shall give examples for each cases.

Note

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

Article information

Source
Nihonkai Math. J., Volume 28, Number 1 (2017), 1-29.

Dates
Received: 24 September 2015
Revised: 14 May 2016
First available in Project Euclid: 7 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1520391677

Mathematical Reviews number (MathSciNet)
MR3771365

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 fringe operator one dimensional perturbation

Citation

Izuchi, Kei Ji; Izuchi, Kou Hei; Izuchi, Yuko. One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk I. Nihonkai Math. J. 28 (2017), no. 1, 1--29. https://projecteuclid.org/euclid.nihmj/1520391677


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