The Reducibility of Truncated Toeplitz Operators
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Complex Analysis and Operator Theory
The Reducibility of Truncated Toeplitz Operators Yufei Li1 · Yixin Yang2
· Yufeng Lu2
Received: 4 December 2019 / Accepted: 18 July 2020 © Springer Nature Switzerland AG 2020
Abstract Let T be a contraction on the Hilbert space 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 AθBn , where Bn is a finite Blaschke product with order n, is given. In particular, we determine when AθBn is reducible in two special cases. One case is that n = 2, 3 and the other case is that Bn = z n (n ∈ N) and θ is a singular inner function. Keywords Reducibility · Truncated Toeplitz operator · Model space Mathematics Subject Classification Primary 47B35 · 47A15; Secondary 47B38
1 Introduction Let D denote the unit disk of the complex plane C, and H 2 denote the Hardy space on D. Note that H 2 is regarded, via non-tangential boundary values on the unit circle T, as a subspace of L 2 (T). For ψ in L ∞ (T), the Toeplitz operator Tψ on H 2 with symbol ψ is defined by
Communicated by Aurelian Gheondea. 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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Yixin Yang [email protected]
Extended author information available on the last page of the article 0123456789().: V,-vol
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Tψ f = P(ψ f ), where P denotes the orthogonal projection on L 2 (T) with range H 2 . To each nonconstant inner function θ , we associate the model space K θ2 = H 2 θ H 2 . The truncated Toeplitz operator on K θ2 with symbol ψ ∈ L ∞ (T) is the operator Aψ defined by Aθψ f = Pθ (ψ f ), where Pθ is the orthogonal projection on L 2 (T) with range K θ2 . Truncated Toeplitz operators on model spaces have been formally introduced by Sarason in [18] and this area has undergone vigorous development during the past several years, see [3,9,10] and references therein. Let T be a bounded linear operator on a Hilbert space H , and a closed subspace M of H is called a reducing subspace of T if T M ⊂ M and T ∗ M ⊂ M. A reducing subspace M is called minimal if the only reducing subspaces contained in M are M and {0}. If T has a proper reducing subspace, we say that T is reducible. The study of invariant subspaces and reducing subspaces for various classes of linear operators has inspired much deep research and prompted many interesting problems. On the Hardy space H 2 , the study of reducing subspaces of multiplication operators probably began with Halmos’s work [14], and it was shown that if φ is an inner function, then the reduc
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