Backward shift invariant subspaces in the bidisc

Izuchi, Keiji; Nakazi, Takahiko

doi:10.14492/hokmj/1285766003

Cited by 39 publications

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“…In general, it is very hard to determine these two functions. Recently, [11] determined these two functions for two well-known invariant subspaces [8,12,13] in H 2 (D 2 ) and they turn out to be strikingly simple. We will focus on the invariant subspace given by the following operatorvalued inner function.…”

Section: Introductionmentioning

confidence: 97%

Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

Yang

2013

Integr. Equ. Oper. Theory

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Section: Introductionmentioning

confidence: 97%

Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

Yang

2013

Integr. Equ. Oper. Theory

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“…For n = 2, the case of f 1 and f 2 are inner was studied in [13,6,7]. Following their method in [7], we characterize M of the form (3.1) in terms of Θ(z 1 ) corresponding to M .…”

Section: Invariant Subspaces Generated By Two Functionsmentioning

confidence: 99%

Two Types of Invariant Subspaces in the Polydisc

Koca

2017

Results Math

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Abstract. It is known that the structure of invariant subspaces of the Hardy space H 2 (D n ) on the polydisc D n is very complicated; hence, we need good examples help us to understand the structure of invariant subspaces of H 2 (D n ). In this paper, we define two types of invariant subspaces of H 2 (D n ). Then, we give a characterization of these types invariant subspaces in view of the Beurling-Lax-Halmos Theorem. Unitary equivalence is also studied in this paper.

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“…It is known [4] that AB| M = V and B A| N = S. Guo and Yang [3] showed that AB is Hilbert-Schmidt under some mild conditions. In this paper, we study M or N when A, B, AB or B A is of finite rank.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we assume that M ⊕ N = H 2 , that is, P M + P N = I where I is the identity operator on H 2 . LetSuppose thatIt is known [4] that AB| M = V and B A| N = S. Guo and Yang [3] showed that AB is Hilbert-Schmidt under some mild conditions. In this paper, we study M or N when A, B, AB or B A is of finite rank.…”

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confidence: 99%

Invariant Subspaces in the Bidisc and Wandering Subspaces

Nakazi¹

2008

JAZ

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Let M be a forward-shift-invariant subspace and N a backward-shift-invariant subspace in the Hardy space H 2 on the bidisc. We assume that H 2 = N ⊕ M. Using the wandering subspace of M and N , we study the relations between M and N . Moreover we study M and N using several natural operators defined by shift operators on H 2 .2000 Mathematics subject classification: primary 47A15, 46J15; secondary 47A20. Keywords and phrases: invariant subspace, two variable Hardy space, wandering subspace. IntroductionLet 2 be the torus, that is, the Cartesian product of two unit circles in C . Let p = 2 or p = ∞. The usual Lebesgue spaces, with respect to the Haar measure m on 2 , are denoted by L p = L p ( 2 ), and H p = H p ( 2 ) is the space of all f in L p whose Fourier coefficientsf ( j, ) = 2 f (z, w)z jw dm (z, w) vanish if at least one of j and is negative. Then H p is called the Hardy space. As 2 = z × w , H p ( z ) and H p ( w ) denote the one-variable Hardy spaces. Let P H 2 be the orthogonal projection from L 2 onto H 2 . For φ in L ∞ , the Toeplitz operator T φ is defined by T * z N ⊂ N and T * w N ⊂ N . Let P M and P N be the orthogonal projections from H 2 onto M and N , respectively. In this paper, we assume that M ⊕ N = H 2 , that is, P M + P N = I where I is the identity operator on H 2 . LetSuppose thatIt is known [4] that AB| M = V and B A| N = S. Guo and Yang [3] showed that AB is Hilbert-Schmidt under some mild conditions. In this paper, we study M or N when A, B, AB or B A is of finite rank. Izuchi and Nakazi For a forward-shift-invariant subspace M, putThese are called wandering subspaces for M. In this paper, [·] denotes the closed span. For a backward-shift-invariant subspace N , with M = H 2 N , putThese are called wandering subspaces for N . In Section 2 we decompose and study M and N using the wandering subspaces M 1 , M 2 , N 1 and N 2 . In Section 3 we study M and N when A or B is of finite rank. For an operator K , r (K ) denotes the rank of K . In Section 4 we show that r (AB) = dim N 1 ∩ N 2 in general, and r (B A) = dim M 1 ∩ M 2 under some mild conditions.In this paper, for a bounded linear operator X on H 2 , ranX = X H 2 and ker Wandering subspacesLet M be a forward-shift-invariant subspace, and N be a backward-shift-invariant subspace with Invariant subspaces in the bidisc and wandering subspaces 369andIn the case of one variable,is an inner function of one variable. THEOREM 1. Let N be a backward-shift-invariant subspace and M = H 2 N .z , then by definition z n f ∈ N for any n ≥ 1, and hence f is orthogonal to [Hence z 2 f ∈ N . By repeating the same argument, we can show that z n f belongs to N for any n ≥ 1. This implies (2). 2 COROLLARY 2. Let N be a backward-shift-invariant subspace. Conversely, if M = q 2 H 2 then M 1 = q 2 H 2 ( w ), and so N 1 = T * z M 1 = 0. Part (2) is clear by (2) of Theorem 1.2By (1) of Theorem 1, both M 1 and M 2 are cyclic subspaces for T z and T w : that is,It may happen that In general, N 0 may not be a cyclic subspace because N 0 = [0] may ...

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Backward shift invariant subspaces in the bidisc

Cited by 39 publications

References 0 publications

Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

Two Types of Invariant Subspaces in the Polydisc

Invariant Subspaces in the Bidisc and Wandering Subspaces

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