Mixed invariant subspaces over the bidisk II
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MIXED INVARIANT SUBSPACES OVER THE BIDISK II Kou Hei Izuchi · Kei Ji Izuchi
Received: 26 May 2011 / Published online: 23 April 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract For a mixed invariant subspace N of H 2 under Tz and Tw∗ , in the previous paper we studied the case rank[Vz , Vw ] = 1 and either dim(N zN ) = 0 or 1. In this paper, we study the structure of N satisfying rank[Vz , Vw ] = 1 and dim(N zN ) = 2. Our study is deeply concerned with the structure of nonextreme points in the closed unit ball of the space of one variable bounded analytic functions. Keywords Invariant subspace · Backward shift invariant subspace · Mixed invariant subspace · Hardy space Mathematics Subject Classification (2000) Primary 47A15 · 32A35 · Secondary 47B35
1 Introduction Let H 2 (z) be the z-variable Hardy space on the open unit disk D. For f (z) ∈ H 2 (z), it is well known that there is the radial limit function f (eiθ ) a.e. on Γ := ∂D. The norm of f is given by 2π 1/2 iθ 2 f e f = dθ/2π . 0 ∞
Let H (z) be the space of bounded functions in H 2 (z). A function f (z) ∈ H ∞ (z) is said to be inner if |f (eiθ )| = 1 a.e. on Γ (see [3]).
K.H. Izuchi Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi 753-8511, Japan e-mail: [email protected]
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K.J. Izuchi ( ) Department of Mathematics, Niigata University, Niigata 950-2181, Japan e-mail: [email protected]
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K.H. IZUCHI, K.J. IZUCHI
Let H 2 be the Hardy space over the bidisk with variables z and w. The norm of F (z, w) ∈ H is given by 2π 2π 1/2 iθ it 2 F e , e dθ dt/(2π)2 . F = 2
0
0
Let Tz and Tw be the operators on H 2 defined by Tz f = zf and Tw f = wf . It is known that the adjoint operator of Tz is given by Tz∗ f = (f − f (0, w))/z. A closed subspace M of H 2 is said to be invariant if Tz M ⊂ M and Tw M ⊂ M. One of main subjects of the study of H 2 is to reveal the structure of invariant subspaces (see [1, 11]). Let Rz , Rw be the compression operators of Tz , Tw on M. In [10], Mandrekar proved that [Rz , Rw∗ ] := Rz Rw∗ − Rw∗ Rz = 0 if and only if M = ϕH 2 for some inner function ϕ. In [4, 5], the authors studied the structure of invariant subspaces M of H 2 satisfying rank[Rz , Rw∗ ] = 1. But it remained some unsolved problems. The difficulty of these problems comes from a lack of function theoretic representation for some functions in H 2 satisfying some kind of properties. Let M ⊥ = H 2 M. Then Tz∗ M ⊥ ⊂ M ⊥ and Tw∗ M ⊥ ⊂ M ⊥ . The space M ⊥ is called a backward shift invariant subspace of H 2 . Let Sz , Sw be the compression operators of Tz , Tw on M ⊥ . In [8], Nakazi, Seto and the first author determined M satisfying [Sz , Sw∗ ] = 0. In [6], the authors determined M satisfying rank[Sz , Sw∗ ] = 1. But we do not know the structure of M satisfying rank[Sz , Sw∗ ] = 2. To shed light on the above problems, in [9] Naito and the authors introduced the concept of “mixed invariant
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