Invertible Composition Operators Research Articles

Composition operators on Hardy spaces of the homogenous rooted trees

In Muthukumar and Ponnusamy (Bull Malays Math Sci Soc 40(4):1801–1815, 2017), the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space \(\mathbb {T}_{p}\) defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator \(C_\phi \) to be bounded on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces. Also, we discuss the compactness of \(C_\phi \) on \({\mathbb {T}}_{p}\) and finally prove there are no compact composition operators on \({\mathbb {T}}_{p,0}\) spaces.

Complex symmetry of invertible composition operators

A bounded operator T on a separable Hilbert space H is said to be complex symmetric if there exists an orthonormal basis for H with respect to which T has a self-transpose matrix representation. In this paper, we study the complex symmetry of composition operators Cϕf=f∘ϕ induced on the Hardy space H2 by holomorphic self-maps ϕ of the open unit disk D. For any holomorphic self-map ϕ of D, we establish that if Cϕ is complex symmetric, then ϕ must fix a point in D. Thus among the automorphisms of D, only the elliptic ones may induce complex symmetric composition operators. For an elliptic automorphism ϕ, we prove that if ϕ is not a rotation or of order 3, then Cϕ is complex symmetric if and only ifϕ(z)=α−z1−α¯z for some α∈D∖{0}.

Isometric and Invertible Composition Operators on Weighted Bergman Spaces of Dirichlet Series

We show that a composition operator on weighted Bergman spaces of Dirichlet series \(\mathcal {A}_{\mu }^p\) is invertible if and only if it is Fredholm if and only if it is an isometry.

Invertible Composition Operators: The Product of a Composition Operator with the Adjoint of a Composition Operator

In this paper, we study the product of a composition operator $$C_{\varphi }$$ with the adjoint of a composition operator $$C^{*}_{\psi }$$ on the Hardy space $$H^2(\mathbb {D})$$ . The order of the product gives rise to two different cases. We completely characterize when the operator $$C_{\varphi }C^{*}_{\psi }$$ is invertible, isometric, and unitary and when the operator $$C^{*}_{\psi }C_{\varphi }$$ is isometric and unitary. If one of the inducing maps $$\varphi $$ or $$\psi $$ is univalent, we completely characterize when $$C^{*}_{\psi }C_{\varphi }$$ is invertible.

On a Localised Single–Valued Extension Property

In this article, we investigate, a certain localised version of the single-valued extension property for a bounded linear operator on a Banach space. We show that this condition behaves canonically under the Riesz functional calculus, and derive a number of characterisations in terms of kernel-type and range-type spaces for the operator and its adjoint. The theory is exemplified in the case of isometries, analytic Toeplitz operators, invertible composition operators on Hardy spaces, and weighted shifts.

Local spectral theory for invertible composition operators onH p

Invertible composition operators on the Hardy spaceHp have automorphic symbols. For 1<p<∞ andp≠2 it is shown that some elliptic composition operators are scalar while others are generalized scalar but not spectral, that parabolic composition operators are generalized scalar but not spectral and that hyperbolic composition operators do not have the single-valued extension property.

A model for invertible composition operators on $H^2$

A model for invertible composition operators on $H^2$

Invertible composition operators on Hp

The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H 2 is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.

Normal and invertible composition operators

Let N denote the set of natural numbers and let φ be a mapping from N into itself. Then the composition transformmation Cφ, on the weighted l2 space with weights a2n, where n ∈ N and 0 &lt; a &lt; 1 is defined by Cφf = f ∘ φ. If Cφ is a bounded operator, then it is called a composition operator. The adjoint of the composition operator Cφ, is computed, and it is used to characterise normal, unitary, isometric, and co-isometric composition operators. Not every invertible φ induces an invertible composition operator, as is shown by examples. At the end of this note all invertible composition operators are characterised.

Invertible Composition Operators on L 2 (λ)

Let C<, be a composition operator on L?(X), where X is a a-finite measure defined on the Borel subsets of a standard Borel space. In this paper a necessary and sufficient condition for the invertibility of C<, is given in terms of invertibility of 0. Also all invertible composition operators on L2(R) induced by monotone continuous functions are characterised. Introduction. Let (X, S, X) be a a-finite measure space, and let ( be a measurable transformation from X into itself. Let L3(X) denote the Hilbert space of all square-integrable functions on X. Define a composition transformation C. on L3(X) as C-f = f o p for every f E L3(X). In case C, is a bounded operator with the range in L3(A), we call it a composition operator induced by 4. The Banach space of all bounded linear operators will be denoted by S. The purpose of this paper is to study the invertible composition operators. The invertible composition operators on HP (of the unit disk) are studied by Schwartz [5], where he proves that the invertibility of the inducing function on the unit disk into itself is a necessary and sufficient condition for the invertibility of the corresponding composition operator. This is true because the inducing functions are analytic functions and hence they are nicely behaved. In case of composition operators on L2(X) the above statement is not completely true as is shown later by an example. However, a little more addition to the hypothesis makes it go through both ways in case of L2(X). 2. An invertibility theorem. DEFINITION. Let (X,S,X) be a measure space. Let 0 be a measurable transformation on X into itself. Then 0 is said to be one-to-one if there exists a measurable transformation 4' on X into itself such that (4p o +)(x) = x a.e. 0 is said to be onto if there exists a measurable transformation X such that (po w) (x) = x a.e. 0 is said to be invertible if there is a measurable transformation 4, such that (0 o A)(x) = (4o +)(x) = x a.e. Such 4 is called the inverse of 0 and is denoted as 0DEFINITION. A standard Borel space X is a Borel subset of a complete separable metric space T. The class S will consist of all the sets of the form X n B, where B is a Borel subset of T. Received by the editors March 21, 1975. AMS (MOS) subject classifications (1970). Primary 47B99; Secondary 47B99.

Invertible composition operators on 𝐿²(𝜆)

Let C ϕ {C_\phi } be a composition operator on L 2 ( λ ) {L^2}(\lambda ) , where λ \lambda is a σ \sigma -finite measure defined on the Borel subsets of a standard Borel space. In this paper a necessary and sufficient condition for the invertibility of C ϕ {C_\phi } is given in terms of invertibility of ϕ \phi . Also all invertible composition operators on L 2 ( R ) {L^2}({\mathbf {R}}) induced by monotone continuous functions are characterised.