Conjugations in $$L^2$$ L 2 and their invariants
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Conjugations in L2 and their invariants M. Cristina Câmara1 · Kamila Kli´s–Garlicka2 · Bartosz Łanucha3 · Marek Ptak2 Received: 3 April 2019 / Accepted: 1 April 2020 © The Author(s) 2020
Abstract Conjugations in space L 2 of the unit circle commuting with multiplication by z or intertwining multiplications by z and z¯ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces. Keywords Conjugation · C–symmetric operator · Hardy space · Invariant subspaces for the unilateral shift · Model space · Truncated Toeplitz operator Mathematics Subject Classification Primary 47B35; Secondary 30D20 · 30H10
The work of the first author was partially supported by FCT/Portugal through UID/MAT/04459/2019 and the research of the second and the fourth authors was financed by the Ministry of Science and Higher Education of the Republic of Poland.
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Kamila Kli´s–Garlicka [email protected] M. Cristina Câmara [email protected] Bartosz Łanucha [email protected] Marek Ptak [email protected]
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Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001 Lisboa, Portugal
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Department of Applied Mathematics, University of Agriculture, ul. Balicka 253c, 30-198 Kraków, Poland
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Department of Mathematics, Maria Curie-Skłodowska University, Maria Curie-Skłodowska Square 1, 20-031 Lublin, Poland 0123456789().: V,-vol
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M. C. Câmara et al.
1 Introduction Let H be a complex Hilbert space and denote by B(H) the algebra of all bounded linear operators on H. A conjugation C in H is an antilinear isometric involution, i.e., C 2 = idH and Cg, Ch = h, g
for g, h ∈ H.
(1.1)
Conjugations have recently been intensively studied and the roots of this subject comes from physics. An operator A ∈ B(H) is called C–symmetric if C AC = A∗ (or equivalently AC = C A∗ ). A strong motivation to study conjugations comes from the study of complex symmetric operators, i.e., those operators that are C–symmetric with respect to some conjugation C. For references see for instance [2,3,7–10]. Hence obtaining the full description of conjugations with certain properties is of great interest. Let T denote the unit circle, and let m be the normalized Lebesgue measure on T. Consider the spaces L 2 = L 2 (T,m), L ∞ = L ∞ (T, m), the classical Hardy space H 2 on the unit disc D identified with a subspace of L 2 , and the Hardy space H ∞ of all analytic and bounded functions in D identified with a subspace of L ∞ . Denote by Mϕ the operator defined on L 2 of multiplication by a function ϕ ∈ L ∞ . The most natural conjugation in L 2 is J defined by J f = f¯, for f ∈ L 2 . This conjugation has two natural properties: the operator Mz is J –symmetric, i.e., Mz J = J Mz¯ , and J maps an analytic function into a co-analytic one, i.e., J H 2 = H 2 . df
Another natural conjugation in L 2 is J f = f # with f # (z) = f (¯z )
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