Conjugations in $$L^2(\mathcal {H})$$ L 2 ( H )
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Integral Equations and Operator Theory
Conjugations in L2(H) M. Cristina Cˆamara, Kamila Kli´s-Garlicka , Bartosz L anucha and Marek Ptak Abstract. Conjugations commuting with Mz and intertwining Mz and Mz¯ in L2 (H), where H is a separable Hilbert space, are characterized. We also investigate which of them leave invariant the whole Hardy space H 2 (H) or a model space KΘ = H 2 (H) ΘH 2 (H), where Θ is a pure operator valued inner function. Mathematics Subject Classification. Primary 47B35; Secondary 47B32, 30D20. Keywords. Conjugation, C-symmetric operator, Hardy space, Model space, Invariant subspaces for unilateral shift, Model for a contraction, Truncated Toeplitz operator.
1. Introduction The motivation to study conjugations (i.e., antilinear isometric involutions) has its roots in physics ([8]), in particular in non-hermitian quantum mechanics and spectral analysis of complex symmetric operators. There are many important examples of complex symmetric operators, that is C-symmetric operators with respect to some conjugation C, namely normal operators, Hankel operators, truncated Toeplitz operators (see for example [3–9,11,14]). In [2,3] all conjugations in the classical L2 space on the unit circle commuting with Mz or intertwining the operators Mz and Mz¯ (in other words, all conjugations C according to which the operator Mz is C-symmetric, see the definition below) were fully characterized. The behaviour of such conjugations was also studied in connection with an analytic part of the space L2 and model spaces, in particular there were characterized all conjugations leaving the whole Hardy space and model spaces invariant. In what follows we study similar questions concerning conjugations in L2 spaces with values in a The work of the first author was partially supported by FCT/Portugal through UID/MAT/04459/2019. 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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certain Hilbert space H. The investigation in this direction is important for its relation with Sz.-Nagy–Foia¸s theory [12, Chap. 6] saying that C0 contractions with finite defect indexes are unitarily equivalent to multiplication by the independent variable in a certain model space given by an operator valued inner function. In other words, the results from the paper can be moved by unitary equivalence to contractions on Hilbert spaces, keeping suitable assumptions. Denote by H a complex Hilbert space, by L(H) the algebra of all bounded linear operators on H and by LA(H) the space of all bounded antilinear operators on H. A conjugation C in H is an antilinear isometric involution, i.e., C 2 = IH and Cf, Cg = g, f
for all f, g ∈ H.
(1.1) ∗
An operator A ∈ L(H) is called C-symmetric if CAC = A . Recall that for A ∈ LA(H) there exists a unique antilinear operator A , called the antilinear adjoint of A, defined by the equality Af, g = f, A g,
(1.2)
for all f, g ∈ H. It is clear, see [
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