The aim of this paper is to investigate asymmetric truncated Toeplitz operators with $L^2$ symbols between two different model spaces given by inner functions such that one divides the other. Characterizations of these operators are given in terms of rank two operators. A description of the class of symbols for which the corresponding asymmetric truncated Toeplitz operator is equal to the zero operator is also given

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'

We show that a weighted shift on a directed tree is related to a multiplier algebra of coefficients of analytic functions. We use this relation to study spectral properties of the operator in question.