The commutant and invariant subspaces for dual truncated Toeplitz operators
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00102-w ORIGINAL PAPER
The commutant and invariant subspaces for dual truncated Toeplitz operators Yongning Li1,2 · Yuanqi Sang3 · Xuanhao Ding1,2 Received: 25 March 2020 / Accepted: 10 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract Dual truncated Toeplitz operators on the orthogonal complement of the model space Ku2 (= H 2 ⊖ uH 2 ) with u nonconstant inner function are defined to be the compression of multiplication operators to the orthogonal complement of Ku2 in L2 . In this paper, we give a complete characterization of the commutant of dual truncated Toeplitz operator Dz , and we even obtain the commutant of all dual truncated Toeplitz operators with bounded analytic symbols. Moreover, we describe the nontrival invariant subspaces of Dz. Keywords Dual truncated Toeplitz operator · Commutant · Invariant subspace · Reducing subspaces Mathematics Subject Classification 47B35 · 47A15 · 42B30
Communicated by Christian Le Merdy. * Yuanqi Sang [email protected] Yongning Li [email protected] Xuanhao Ding [email protected] 1
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, People’s Republic of China
2
Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing 400067, People’s Republic of China
3
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, People’s Republic of China
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1 Introduction Let H 2 be the classical Hardy space of open unit disk 𝔻 = {z ∈ ℂ ∶ |z| < 1} and L2 = L2 (𝕋 ) denote the usual Lebesgue space on the unit circle 𝕋 = {z ∈ ℂ ∶ |z| = 1} . Let H ∞ be the space of bounded analytic functions on 𝔻 . By Fatou’s theorem and harmonic extension [12], we usually identify H 2 with the closed subspace of L2 (𝕋 ) consisting of the functions with vanishing fourier coefficients with negative indices. Let P denote the orthogonal projection from L2 to H 2 . For 𝜑 ∈ L∞ , the Toeplitz operator T𝜑 ∶ H 2 → H 2 is defined by
T𝜑 x = P(𝜑x), x ∈ H 2 . Let P− = I − P denote the orthogonal projection from L2 to (H 2 )⊥ = zH 2 , the Hankel operator H𝜑 ∶ H 2 → (H 2 )⊥ is defined by
H𝜑 x = P− (𝜑x), x ∈ H 2 and an easy calculation gives
H𝜑∗ y = P(𝜑y), ̄ y ∈ (H 2 )⊥ . The dual Toeplitz operator S𝜑 defined on (H 2 )⊥ is given by
S𝜑 x = P− (𝜑x), x ∈ (H 2 )⊥ . For f ∈ L2 , define Vf (z) = z̄ f (z) ; then V is an anti-unitary operator on L2 and satisfies
V = V −1 , VTf = Sf̄ V. Let u be a nonconstant inner function. Ku2 = H 2 ⊖ uH 2 is called the model space, which is an invariant subspace for Tz̄ [2]. Let Pu denote the orthogonal projection from L2 to Ku2 . For 𝜑 ∈ L2 , the truncated Toeplitz operator A𝜑 , which was first introduced by Sarason [19], is defined on Ku2 by
A𝜑 x = Pu (𝜑x), x ∈ Ku2 . The truncated Hankel operator B𝜑 ∶ Ku2 → (Ku2 )⊥ is defined by
B𝜑 x = (I − Pu )(𝜑x), x ∈ Ku2 , and it is easy to check that
B∗𝜑 y = Pu (𝜑x), ̄ y ∈ (Ku2 )⊥ . The dual truncated Toeplitz opera
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