Ranks of Cross-Commutators and Unitary Module Maps
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Ranks of Cross-Commutators and Unitary Module Maps Kei Ji Izuchi1 · Kou Hei Izuchi2 · Yuko Izuchi3
Received: 7 February 2017 / Revised: 12 December 2017 / Accepted: 7 January 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract Let M be an invariant subspace of H 2 over the bi-disk and N = H 2 M. Let Sz,N , Sw,N be the compression of the multiplication operators Tz , Tw on H 2 onto N . For a two-variable inner function θ , let Mθ = θ M and Nθ = H 2 Mθ . We shall study ∗ ] and [S ∗ the relationship of the ranks of the cross-commutators [Sz,N , Sw,N z,Nθ , Sw,Nθ ]. ∗ ∗ We also characterize M such that rank [Sz,N , Sw,N ] = rank [Sz,Nθ , Sw,Nθ ] for any nonconstant inner function θ . Keywords Hardy space over the bi-disk · Invariant subspace · Backward shift invariant subspace · Unitary module map · Rank of cross-commutator Mathematics Subject Classification Primary 47A15 · 32A35; Secondary 47B35
Communicated by Raymond Mortini. The Kei Ji Izuchi is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science (No. 15K04895).
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Kei Ji Izuchi [email protected] Kou Hei Izuchi [email protected] Yuko Izuchi [email protected]
1
Department of Mathematics, Niigata University, Niigata 950-2181, Japan
2
Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi 753-8511, Japan
3
Asahidori 2-2-23, Yamaguchi 753-0051, Japan
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1 Introduction Let H 2 (D) be the classical Hardy space over the open unit disk D and L 2 (∂D) be the Lebesgue space on ∂D. Identifying a function f in H 2 (D) with its radial limit function, limr →1 f (r eit ) for eit ∈ ∂D, we also think of H 2 (D) as a closed subspace of L 2 (∂D). A function ϕ ∈ H 2 (D) is called inner if |ϕ| = 1 a.e. on ∂D. For a bounded measurable function ψ on ∂D, we denote by Tψ the Toeplitz operator on H 2 (D), that is, Tψ f = PH 2 (D) ψ f for f ∈ H 2 (D), where PH 2 (D) is the orthogonal projection from L 2 (∂D) onto H 2 (D). A non-zero closed subspace I of H 2 (D) is said to be invariant if Tz I ⊂ I . The famous Beurling theorem (see [4]) says that there is an inner function ϕ such that I = ϕ H 2 (D), so the structure of the family of invariant subspaces of H 2 (D) is well understood. See [4] for function theory on D. Let H 2 := H 2 (D2 ) be the Hardy space over the bi-disk D2 with variables z, w and L 2 := L 2 (∂D2 ), where ∂D2 = ∂D × ∂D. Identifying a function f in H 2 with limr →1 f (r eit1 , r eit2 ) for (eit1 , eit2 ) ∈ ∂D2 , we also consider that H 2 ⊂ L 2 . A function ϕ ∈ H 2 is also called inner if |ϕ| = 1 a.e. on ∂D2 . A non-zero closed subspace M of H 2 is said to be invariant if Tz M ⊂ M and Tw M ⊂ M. It is known that the structure of the family of invariant subspaces of H 2 is fairly complicated (see [2,10,12–14]). We write Rz,M = Tz | M and Rw,M = Tw | M . Then Rz,M , Rw,M are bounded linear operators on M. In [8], Mandrekar showed that ∗ ∗ ∗ ] := Rz,M Rw,M − Rw,M Rz,M = 0 [Rz,M , Rw,M
if and only if M is Beurling type, that
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