Abstract
In this paper, using some properties about Toeplitz kernels, we present some results about finite-rank properties of the commutator \([A_f,~A_g]\). Firstly, we show that \([A_{B_n},~A_v^*]\) must have a finite rank on the model space \(K_u^2\), where \(B_n\) is a finite Blaschke product and v is an inner function. Next, we present that when \(\text {ker}~T_{\overline{u}B_n}\) is an invariant subspace of \(T_\phi ^*\), then \([A_{B_n},~A_\phi ^*]\) has a finite rank on \(K_u^2\) for \(\phi \in H^\infty \). Finally, we prove that \([A_{B_n},~A_\phi ^*]\) must have a finite rank on \(K_u^2\) when \(u=B_nu_1\) for an inner function \(u_1\).
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This research is supported by National Natural Science Foundation of China No. 12031002, 11671065.
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Communicated by Mohammad Sal Moslehian.
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Yang, X., Lu, Y. Toeplitz Kernels and Finite-Rank Commutators of Truncated Toeplitz Operators. Bull. Malays. Math. Sci. Soc. 45, 2175–2193 (2022). https://doi.org/10.1007/s40840-022-01374-1
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DOI: https://doi.org/10.1007/s40840-022-01374-1