 

Kei Ji Izuchi, Kou Hei Izuchi, and Yuko Izuchi
Wandering subspaces and the Beurling type theorem. II view print


Published: 
November 22, 2010 
Keywords: 
Beurling type theorem, Hardy space over the bidisk, invariant subspace, wandering subspace, fringe operator 
Subject: 
Primary 47A15, 32A35; Secondary 47B35. 


Abstract
Let H^{2} be the Hardy space over the bidisk.
Let φ(w) be a nonconstant inner function.
We denote by [zφ(w)] the smallest invariant subspace for
both operators T_{z} and T_{w} containing the function zφ(w).
Aleman, Richter and Sundberg showed that
the Beurling type theorem holds for the Bergman shift on the
Bergman space. It is known that the compression operator
S_{z} on H^{2}⊝[zw] is unitarily equivalent to the Bergman shift,
so the Beurling type theorem holds for S_{z} on H^{2}⊝[zw].
As a generalization, we shall show that
the Beurling type theorem holds for S_{z} on H^{2}⊝[zφ(w)].
Also we shall prove that the Beurling type theorem holds for the fringe operator
F_{w} on [zw]⊝z[zw] and for F_{z} on [zφ(w)]⊝w[zφ(w)]
if φ(0)=0.


Acknowledgements
The first author is partially supported by GrantinAid for Scientific Research (No.21540166), Japan Society for the Promotion of Science.


Author information
Kei Ji Izuchi:
Department of Mathematics, Niigata University, Niigata 9502181, Japan
izuchi@m.sc.niigatau.ac.jp
Kou Hei Izuchi:
Department of Mathematics, Korea University, Seoul 136701, Korea
kh.izuchi@gmail.com
Yuko Izuchi:
Aoyamashinmachi 186301, Niigata 9502006, Japan
yfd10198@nifty.com

