Conjugations in $$L^2$$ and their invariants

Abstract

Conjugations in space L 2 of the unit circle commuting with multiplication by z or intertwining multiplications by z andz are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces.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][8][9][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 M ϕ 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 M z is J -symmetric, i.e., M z J = J Mz, and J maps an analytic function into a co-analytic one, i.e., J H 2 = H 2 . Another natural conjugation inOpen Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article'

Compressions of Multiplication Operators and Their Characterizations

“…There is a natural conjugation (an antiunitary involution) connected with a model space (see for instance [5,12]). For an inner function θ define…”

“…Model spaces, which provide the natural setting for truncated Toeplitz operators, have generated enormous interest and they are relevant in connection with a variety of topics such as the Schrödinger operator, classical extremal problems in control theory, Hankel operators and Toeplitz matrices (see for instance [13] and [11]). Natural conjugations, which model spaces and the whole L 2 possess (see [5]), make model spaces even more natural in the context of pysics [12]. Their orthogonal complements in L 2 also appear in numerous applications.…”

Conjugations in $$L^2(\mathcal {H})$$

“…In [2,Theorem 2.4] all conjugations in L 2 commuting with M z were characterized. In particular it was shown that such a conjugation has to be of the form M ψ J for some unimodular function ψ ∈ L ∞ which is symmetric, i.e., ψ(z) = ψ(z) a.e.…”

“…In [2,3] all conjugations in the classical L 2 space on the unit circle commuting with M z or intertwining the operators M z and Mz (in other words, all conjugations C according to which the operator M z 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 L 2 and model spaces, in particular there were characterized all conjugations leaving the whole Hardy space and model spaces invariant.…”

Conjugations on Banach $$*$$-algebras

“…E.g. all conjugations in L 2 space of the unit circle commuting with M z or intertwining operators M z and M̄z (in other words all conjugations C according to which the operator M z is C-symmetric, see the definition below) are completely characterized in [3,5]. In [4] similar description is given for L 2 spaces of vector valued functions.…”