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2017 One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk I
Kei Ji Izuchi, Kou Hei Izuchi, Yuko Izuchi
Nihonkai Math. J. 28(1): 1-29 (2017).

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.

Funding Statement

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

Citation

Download Citation

Kei Ji Izuchi. Kou Hei Izuchi. Yuko Izuchi. "One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk I." Nihonkai Math. J. 28 (1) 1 - 29, 2017.

Information

Received: 24 September 2015; Revised: 14 May 2016; Published: 2017
First available in Project Euclid: 7 March 2018

zbMATH: 06714331
MathSciNet: MR3771365

Subjects:
Primary: 32A35‎ , 47A15
Secondary: 47B35

Keywords: fringe operator , Hardy space over the bidisk , invariant subspace , one dimensional perturbation

Rights: Copyright © 2017 Niigata University, Department of Mathematics

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Vol.28 • No. 1 • 2017
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