## Nihonkai Mathematical Journal

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

#### 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
Revised: 14 May 2016
First available in Project Euclid: 7 March 2018

https://projecteuclid.org/euclid.nihmj/1520391677

Mathematical Reviews number (MathSciNet)
MR3771365

Zentralblatt MATH identifier
06714331

#### 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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