Ranks of commutators for a class of truncated Toeplitz operators
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00104-8 ORIGINAL PAPER
Ranks of commutators for a class of truncated Toeplitz operators Yong Chen1 · Young Joo Lee2 · Yile Zhao1 Received: 13 June 2020 / Accepted: 14 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract We consider truncated Toeplitz operators acting on infinite dimensional model spaces. We then describe the kernels and ranks of commutators of truncated Toeplitz operators with symbols induced by certain inner functions. Our results generalize recent results of Chen et al. [Oper Matrices (to appear)] to infinite dimensional model spaces. Keywords Truncated Toeplitz operator · Model space · Rank Mathematics Subject Classification 47B35 · 32A37
1 Introduction Let 𝔻 be the unit disk in the complex place and 𝕋 be the unit circle. We let H 2 be the classical Hardy space on 𝔻 which can be identified with a closed subspace of L2 . Here, Lp ∶= Lp (𝕋 , 𝜎) denotes the usual Lebesgue space on 𝕋 where 𝜎 is the normalized Lebesgue measure on 𝕋 . A function 𝜃 ∈ H 2 is said to be inner if |𝜃| = 1 a.e. on 𝕋 . To each non-constant inner function 𝜃 , we associate the model space H𝜃 defined by
Communicated by Jan Stochel. * Young Joo Lee [email protected] Yong Chen [email protected] Yile Zhao [email protected] 1
Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, People’s Republic of China
2
Department of Mathematics, Chonnam National University, Gwangju 61186, South Korea
Vol.:(0123456789)
Y. Chen et al.
H𝜃 = H 2 ⊖ 𝜃H 2 which is a nontrivial invariant subspace for the backward shift operator on H 2 . Let P𝜃 be the Hilbert space orthogonal projection from L2 to H𝜃 . Given a function 𝜑 ∈ L∞ , the truncated Toeplitz operator (briefly, TTO) A𝜑 with symbol 𝜑 is defined on H𝜃 by
A𝜑 f = P𝜃 (𝜑f ) for functions f ∈ H𝜃 . Then A𝜑 is a bounded linear operator on H𝜃 and A∗𝜑 = A𝜑 ; see [9] for details and related facts. In a recent paper [4], the rank of a commutator of two TTOs has been studied and it has been shown that for any 𝜑, 𝜓 ∈ L∞ and inner 𝜃 , the rank of [A𝜑 , A𝜓 ] must be even on H𝜃 if it has finite rank. Here, [S, T] ∶= ST − TS denotes the commutator of two bounded linear operators S and T on a Hilbert space. Conversely, it has been also studied that whether there is a commutator of two TTOs with a given even rank exactly on a model space. At the same paper, it was proved that this is true on model spaces corresponding to monomials by showing the selfcommutator [AzN , A∗zN ] has rank exactly 2N on H(−z)n when 2N ≤ n ; see Proposition 7 of [4]. Motivated by this result, one might naturally ask whether the same is true on any infinite dimensional model spaces corresponding to general inner functions. More generally, one can naturally ask the following question. Question. For two inner functions 𝜃 and 𝜂 , what is the rank of the self-commutator [A𝜂 , A∗𝜂 ] on H𝜃? Recently, this question has been studied for finite dimensional model spaces corresponding to finite Blaschke products. To introduce the result
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