Compressions of Multiplication Operators and Their Characterizations
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Results in Mathematics
Compressions of Multiplication Operators and Their Characterizations M. Cristina Cˆamara, Kamila Kli´s–Garlicka, Bartosz L anucha , and Marek Ptak Abstract. Dual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable Mz on the classical L2 space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated. Mathematics Subject Classification. 47B32, 47B35, 30H10. Keywords. Model space, truncated Toeplitz operator, dual truncated Toeplitz operator.
1. Introduction Let L2 := L2 (∂D) be the space of all measurable and square integrable functions on the unit circle T with respect to the normalized Lebesgue’s measure. Recall that for ϕ ∈ L∞ the multiplication operator Mϕ is defined by Mϕ f = 2 = L2 H 2 , let P + ϕf , f ∈ L2 . Let H 2 denote the classical Hardy space and H− 2 2 stand for the orthogonal projection from L onto H and let P − = IL2 − P + 2 be the orthogonal projection from L2 onto H− . For ϕ ∈ L∞ recall the standard definitions Tϕ = P + Mϕ|H 2 ,
Hϕ = P − Mϕ|H 2 .
The work of the first author was partially supported by FCT/Portugal through UID/MAT/04459/2020. The research of the second and the fourth authors was financed by the Ministry of Science and Higher Education of the Republic of Poland. 0123456789().: V,-vol
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2 The operators: Tϕ ∈ B(H 2 ) and Hϕ ∈ B(H 2 , H− ) are called Toeplitz operator and Hankel operator (with symbol ϕ), respectively. We will use the notation B(H ) or B(H , K ) for the set of all bounded operators on the Hilbert space H or from H to K . In what follows let θ denote a nonconstant inner function. Recall that the model space Kθ is defined as the orthogonal complement of θH 2 in H 2 . One can therefore consider the decompositions
H 2 = Kθ ⊕ θH 2
and
2 L2 = Kθ ⊕ (Kθ )⊥ = Kθ ⊕ θH 2 ⊕ H− .
According to the decomposition L2 = Kθ ⊕ (Kθ )⊥ the operator Mϕ , ϕ ∈ L∞ , can be written as θ Aϕ (Bϕθ¯ )∗ Mϕ = . (1.1) Bϕθ Dϕθ Although Mϕ is bounded if and only if ϕ ∈ L∞ , some restrictions of Mϕ may still be bounded even if they do not have a bounded symbol (for example, for Aθϕ see [1] and for Bϕθ see Remark 4). Hence, now let ϕ ∈ L2 and set the densely defined multiplication operator Mϕ : D(Mϕ ) → L2 as Mϕ f = ϕf , where D(Mϕ ) = {f ∈ L2 : ϕf ∈ L2 }. Note that L∞ ⊂ D(Mϕ ) for all ϕ ∈ L2 . Let Pθ denote the orthogonal projection from L2 onto Kθ and let Pθ⊥ = IL2 − Pθ be the orthogonal projection from L2 onto (Kθ )⊥ . Recall after [11] that Kθ∞ := Kθ ∩ L∞ is a dense subset of Kθ . Since z¯H ∞ is a dense subset 2 and θH ∞ is a dense subset of θH 2 , it follows that Kθ⊥ ∩ L∞ is a dense of H− subset of Kθ⊥ . We define Aθϕ = Pθ Mϕ|Kθ ∩L∞ ,
Bϕθ = Pθ⊥ Mϕ|Kθ ∩L∞
and
Dϕθ = Pθ⊥ Mϕ|Kθ⊥ ∩L∞ .
If Aθϕ extends to the whole Kθ as a bounded operator, it is called a truncated Toeplitz op
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