Abstract
Let T be a contraction on the Hilbert space \(\mathscr {H}\) and S a minimal isometric dilation of T. In this paper, we show that every projection in \(\{T\}'\) can be extended to a projection in \(\{S\}'\). Using this result, a sufficient condition for reducibility of \(A^{\theta }_{B_{n}}\), where \(B_{n}\) is a finite Blaschke product with order n, is given. In particular, we determine when \(A^{\theta }_{B_{n}}\) is reducible in two special cases. One case is that \(n=2,3\) and the other case is that \(B_{n}=z^{n}\) (\(n\in \mathbb {N}\)) and \(\theta \) is a singular inner function.
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Acknowledgements
We are grateful to the anonymous reviewer for giving us many useful comments and suggestions to improve considerably the earlier version of the paper, especially for simplifying the earlier proofs of Proposition 1.3 and the claim in the proof of Theorem 4.3. The data that support the findings of this study are included in this article.
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Communicated by Aurelian Gheondea.
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This research is supported by National Natural Science Foundation of China (No. 11671065,11971086).
This first author was supported by the Fundamental Research Funds for the Central Universities 2412020QD023, and the second author was partially supported by DUT Fundamental Research Funds for the Central Universities DUT19LK53.
This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.
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Li, Y., Yang, Y. & Lu, Y. The Reducibility of Truncated Toeplitz Operators. Complex Anal. Oper. Theory 14, 60 (2020). https://doi.org/10.1007/s11785-020-01017-y
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DOI: https://doi.org/10.1007/s11785-020-01017-y