scholarly journals Composition Operators on Some Banach Spaces of Harmonic Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Zygmund Space ◽  
Spaces Of Analytic Functions

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.

ON THE HARMONIC ZYGMUND SPACES

2020 ◽  
Vol 101 (3) ◽  
pp. 466-476
Author(s):  
MUNIRAH ALJUAID ◽  
FLAVIA COLONNA
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Zygmund Space ◽  
Class A ◽  
Analytic Case ◽  
Separable Subspace ◽  
Zygmund Spaces

In this paper we study a class ${\mathcal{Z}}_{H}$ of harmonic mappings on the open unit disk $\mathbb{D}$ in the complex plane that is an extension of the classical (analytic) Zygmund space. We extend to the elements of this class a characterisation that is valid in the analytic case. We also provide a similar result for a closed separable subspace of ${\mathcal{Z}}_{H}$ which we call the little harmonic Zygmund space.

On Algebras Generated by Composition Operators

1974 ◽  
Vol 26 (5) ◽  
pp. 1234-1241 ◽  
Author(s):  
J. A. Cima ◽  
W. R. Wogen
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Group Of Automorphisms ◽  
Image Position

Let Δ be the open unit disk in the complex plane and let be the group of automorphisms of Δ onto Δ, define byThe Banach spaces Hp = Hp(Δ), 1 ≦ p < ∞, are the Hardy spaces of functions analytic in Δ with their integral p means bounded,

Generalized Composition Operators on Weighted Hilbert Spaces of Analytic Functions

2017 ◽  
Vol 10 ◽  
pp. 1-13 ◽  
Author(s):  
Waleed Al-Rawashdeh
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Carleson Measures ◽  
Open Unit ◽  
Open Unit Disk ◽  
Admissible Weight ◽  
Spaces Of Analytic Functions

Letφbe an analytic self-map of the open unit disk D andgbe an analytic function on D. The generalized composition operator induced by the mapsgandφis defined by the integral operatorI(g,φ)f(z) =∫0zf′(φ(ς))g(ς)dς. Given an admissible weightω, the weighted Hilbert spaceHωconsists of all analytic functionsfsuch that ∥f∥2Hω= |f(0)|2+∫D|f′(z)|2ω(z)dA(z) is finite. In this paper, we characterize the boundedness and compactness of the generalized composition operators on the spaceHωusing theω-Carleson measures. Moreover, we give a lower bound for the essential norm of these operators.

scholarly journals Fredholm Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions of TypeH∞

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
M. Carmen Gómez-Collado ◽  
David Jornet
Keyword(s):  
Banach Spaces ◽  
Analytic Functions ◽  
Unit Disc ◽  
Composition Operators ◽  
Open Unit ◽  
Weighted Composition Operators ◽  
Open Unit Disc ◽  
Weighted Composition ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

We study Fredholm (weighted) composition operators between general weighted Banach spaces of analytic functions on the open unit disc with sup-norms.

scholarly journals Planar harmonic mappings in a family of functions convex in one direction

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 431-445
Author(s):  
Sudhananda Maharan ◽  
Swadesh Sahoo
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Harmonic Mapping ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Optimal Estimates ◽  
Convex In One Direction ◽  
Planar Harmonic Mappings ◽  
Class Of Functions

Let D := {z ? C : |z| < 1} be the open unit disk, and h and 1 be two analytic functions in D. Suppose that f = h + ?g is a harmonic mapping in D with the usual normalization h(0) = 0 = g(0) and h'(0) = 1. In this paper, we consider harmonic mappings f by restricting its analytic part to a family of functions convex in one direction and, in particular, starlike. Some sharp and optimal estimates for coefficient bounds, growth, covering and area bounds are investigated for the class of functions under consideration. Also, we obtain optimal radii of fully convexity, fully starlikeness, uniformly convexity, and uniformly starlikeness of functions belonging to those family.

scholarly journals On a Class of Composition Operators on Bergman Space

2007 ◽  
Vol 2007 ◽  
pp. 1-11
Author(s):  
Namita Das ◽  
R. P. Lal ◽  
C. K. Mohapatra
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Measurable Function ◽  
Analytic Functions ◽  
Composition Operators ◽  
Linear Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Space Of Analytic Functions ◽  
Bounded Linear

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

scholarly journals WEIGHTED COMPOSITION OPERATORS ON H∞ ∩ o

2014 ◽  
Vol 57 (2) ◽  
pp. 475-480
Author(s):  
SHÛICHI OHNO
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Composition Operators ◽  
Open Unit ◽  
Weighted Composition Operators ◽  
Open Unit Disk ◽  
Bounded Analytic Functions ◽  
Disk Algebra ◽  
Closed Subalgebra ◽  
Weighted Composition

AbstractWe will characterize the boundedness and compactness of weighted composition operators on the closed subalgebra H∞ ∩ $\mathcal{B}$o between the disk algebra and the space of bounded analytic functions on the open unit disk.

scholarly journals DIFFERENCES OF COMPOSITION OPERATORS BETWEEN WEIGHTED BERGMAN SPACES AND WEIGHTED BANACH SPACES OF HOLOMORPHIC FUNCTIONS

2010 ◽  
Vol 52 (2) ◽  
pp. 325-332 ◽  
Author(s):  
ELKE WOLF
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Holomorphic Functions ◽  
Weighted Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
Spaces Of Holomorphic Functions ◽  
Weighted Banach Spaces

AbstractWe characterise boundedness and compactness of differences of composition operators acting between weighted Bergman spaces Av, p and weighted Banach spaces H∞w of holomorphic functions defined on the open unit disk D.

scholarly journals Characterisation of the isometric composition operators on the Bloch space

2005 ◽  
Vol 72 (2) ◽  
pp. 283-290 ◽  
Author(s):  
Flavia Colonna
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Composition Operator ◽  
Blaschke Product ◽  
Analytic Functions ◽  
Composition Operators ◽  
Bloch Space ◽  
Open Unit ◽  
Open Unit Disk ◽  
Identity Function

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

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