Class Of Composition Operators Research Articles

On Sharp Estimates of Coefficients for Polynomials in Norm-Attainable Classes

New sharp estimates for coefficients of pseudo-monoidal type polynomials are given for the class of norm-attainable operators. Moreover, we show the existence of an equivalence relation in the case of the class of norm-attainable operators with the Bergman determinant for the class of composition operators.

Composition operators on Bloch and Hardy type spaces

The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.

A Review on Classes of Composition Operators

A Review on Classes of Composition Operators

Rigidity of composition operators on the Hardy space Hp

Let ϕ be an analytic map taking the unit disk D into itself. We establish that the class of composition operators f↦Cϕ(f)=f∘ϕ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space Hp, for 1≤p<∞ and p≠2. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) Cϕ is a compact operator Hp→Hp, (ii) Cϕ fixes a (linearly isomorphic) copy of ℓp in Hp, but Cϕ does not fix any copies of ℓ2 in Hp, (iii) Cϕ fixes a copy of ℓ2 in Hp. Moreover, in case (iii) the operator Cϕ actually fixes a copy of Lp(0,1) in Hp provided p>1. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on Hp, which contain the compact operators K(Hp). In particular, the class of composition operators on Hp does not reflect the quite complicated lattice structure of such ideals.

Quasi-parabolic Composition Operators on Weighted Bergman Spaces

In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_{\alpha}^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $\varphi(z)=$ $z+\psi(z)$, where $\psi\in$ $H^{\infty}(\mathbb{H})$ and $\Im(\psi(z)) > \epsilon > 0$. We especially examine the case where $\psi$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

The single equality [formula omitted] does not imply the quasinormality of weighted shifts on rootless directed trees

It is proved that each bounded injective bilateral weighted shift W satisfying the equality W⁎nWn=(W⁎W)n for some integer n⩾2 is quasinormal. For any integer n⩾2, an example of a bounded non-quasinormal weighted shift A on a rootless directed tree with one branching vertex which satisfies the equality A⁎nAn=(A⁎A)n is constructed. It is also shown that such an example can be constructed in the class of composition operators in L2-spaces over σ-finite measure spaces.

Aluthge transforms of weighted shifts on directed trees

Aluthge transform of a bounded operator is generalized to the case of unbounded one. A formula for the Aluthge transform of a weighted shift on a directed tree is established and it is used to construct an example of a hyponormal operator whose Aluthge transform has trivial domain. It is proven that such an example can be also constructed in the class of composition operators. It is also shown that Aluthge transform of a closed, densely defined operator is not necessarily closable.

Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach’s spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.

Localization results for generalized Baskakov/Mastroianni and composite operators

In this paper we study localization results for classical sequences of linear positive operators that are particular cases of the generalized Baskakov/Mastroianni operators and also for certain class of composite operators that can be derived from them by means of a suitable transformation. Amongst these composite operators we can find classical sequences like the Meyer–König and Zeller operators and the Bleimann, Butzer and Hahn ones. We extend in different senses the traditional form of the localization results that we find in the classical literature and we show several examples of sequences with different behavior to this respect.

Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on H2 with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form φ(z)=z+ψ(z), where ψ∈H∞(H) and ℑ(ψ(z))>ϵ>0. We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

ORBITAL LIMIT POINTS AND HYPERCYCLICITY OF OPERATORS ON ANALYTIC FUNCTION SPACES

On a separable, infinite dimensional Banach space X, a bounded linear operator T : X → X is said to be hypercyclic if there exists a vector x in X such that its orbit Orb(T, x)= {x, T x, T 2 x, . . .} is dense in X. It is known that for a weighted backward shift T to be hypercyclic it suffices to merely require the operator to have an orbit Orb(T, x) with a non-zero limit point. In this paper we show that while this result also holds true for the class of adjoints of multiplication operators on the Bergman spaces, it breaks down for the class of composition operators on the Hardy space.

Division by inner functions in a class of composition operators on Lipschitz spaces

We prove that the class 𝒞 ( H Λ α , H Λ β p ) = { ϕ : C ϕ ( H Λ α ) ⊂ H Λ β p } , where HΛα and H Λ β p are holomorphic Lipschitz spaces, has the f-property for p⩾1, 0<β<1: If ϕ I ∈ 𝒞 ( H Λ α , H Λ β p ) where I is an inner function, then ϕ ∈ 𝒞 ( H Λ α , H Λ β p ) Also we characterize inner functions that belong to 𝒞 ( H Λ α , H Λ β p ) for p>0, 0<α⩽1 and 0<β⩽1. The results are deduced from a new, derivative-free, characterization of 𝒞 ( H Λ α , H Λ β p ) that extends some results of Dyakonov [Acta Math. 178 (1997) 143–167.] and the author [Acta Math. 183 (1999) 141–143; J. Math. Anal. Appl., 326 (2007) 1–11].

On a Class of Composition Operators on Bergman Space

Let&amp;#x1D53B;={z∈ℂ:|z|&lt;1}be the open unit disk in the complex planeℂ. LetA2(&amp;#x1D53B;)be the space of analytic functions on&amp;#x1D53B;square integrable with respect to the measuredA(z)=(1/π)dx dy. Givena∈&amp;#x1D53B;andfany measurable function on&amp;#x1D53B;, we define the functionCafbyCaf(z)=f(ϕa(z)), whereϕa∈Aut(&amp;#x1D53B;). The mapCais a composition operator onL2(&amp;#x1D53B;,dA)andA2(&amp;#x1D53B;)for alla∈&amp;#x1D53B;. Letℒ(A2(&amp;#x1D53B;))be the space of all bounded linear operators fromA2(&amp;#x1D53B;)into itself. In this article, we have shown thatCaSCa=Sfor alla∈&amp;#x1D53B;if and only if∫&amp;#x1D53B;S˜(ϕa(z))dA(a)=S˜(z), whereS∈ℒ(A2(&amp;#x1D53B;))andS˜is the Berezin symbol of S.

Compact Multiplication Operators on Weighted Spaces of Vector-Valued Continuous Functions

In this note we characterize the compact multiplication operators Mπ on the weighted locally convex spaces CV0(X,E) of vector-valued continuous functions induced by the operator-valued mappings π : X → B(E). 0. Introduction. Let L(X,E) be a vector space of functions from a nonempty set X to a vector space E over the field C or R. Let T (E) be a set of linear transformations from E to itself. If φ : X → X and π : X → T (E) are mappings such that the weighted composite function π.f ◦ φ belongs to L(X,E), whenever f ∈ L(X,E), then the mapping taking f to π.f ◦φ is a linear transformation on L(X,E) and we denote it by Wπ,φ. In case L(X,E) is a topological vector space and the mapping Wπ,φ is continuous, we call Wπ,φ the weighted composition operator on L(X,E) induced by the symbol (π, φ). In case φ is the identity map, we call it the multiplication operator induced by π and we denote it by Mπ. For details on these operators we refer to [11]. The compact weighted composition operators on spaces of continuous functions have been studied extensively by many authors like Kamowitz [5], Feldman [3], Singh and Summers [13], Jamison and Rajagopalan [4], Takagi [14], Chan [2] and Singh and Manhas [12]. As we know, the class of weighted composition operators include the class of multiplication operators and the class of composition operators. One natural question arises: is it possible to get the behavior of the compact multiplication operators from the study of the compact weighted composition operators? In general, it is not possible since the conditions obtained earlier for a weighted composition operator to be compact is not satisfied by the identity map φ. So it motivates us to 1991 AMS Mathematics Subject Classification. Primary 47B38, 47B07, Secondary 47A56, 46E10, 46E40.

High-gradient operators in the N-vector model

It has been shown by several authors that a certain class of composite operators with many fields and gradients endangers the stability of non-trivial fixed points in 2 + ϵ expansions for various models. This problem is up to now unresolved. We investigate it in the N-vector model in an 1/ N expansion. By establishing an asymptotic naive addition law for anomalous dimensions we demonstrate that the first orders in the 2 + ϵ expansion can lead to erroneous interpretations for high-gradient operators. While this makes us cautious to over-interpret such expansions (either 2 + ϵ or 1/ N), the stability problem in the N-vector model persists also in first order in 1/ N below three dimensions.

A Characterization of a Class of Composition Operators

A Characterization of a Class of Composition Operators

A Characterization of a Class of Composition Operators

A Characterization of a Class of Composition Operators

A new class of composition operators

A new class of composition operators Pϕ : H2(T) → H2(T), with is introduced. Sufficient conditions on ϕ for Pϕ to be bounded and Hilbert‐Schmidt are obtained. Properties of Pϕ with ϕ(eit) = aeit + be−it for different values of the parameters a and b have been investigated. This paper concludes with a discussion on the compactness of Pϕ.