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The Product of Automorphic Weighted Composition Operators on Hardy Space H <sup>2</sup>
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Abstract<p>Let <inline-formula> <tex-math><?CDATA $n\in {\mathbb{N}},{p}_{i}\in {\rm{U}},{\alpha }_{{P}_{i}}(z)=\frac{{p}_{i}-z}{1-{\bar{p}}_{i}z}(z\in {\rm{U}})$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <mi>n</mi> <mo>∈</mo> <mi>ℕ</mi> <mo>,</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi mathvariant="normal">U</mi> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>−</mo> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mover accent="true"> <mi>p</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> <mi>z</mi> </mrow> </mfrac> <mo stretchy="false">(</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="normal">U</mi> <mo stretchy="false">)</mo> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn1.gif" xlink:type="simple"></inline-graphic> </inline-formula>, and let <inline-formula> <tex-math><?CDATA ${f}_{1}\in {H}^{\infty },i=1,\ldots,n$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>∈</mo> <msup> <mi>H</mi> <mi>∞</mi> </msup> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn2.gif" xlink:type="simple"></inline-graphic> </inline-formula>. We discuss the relation between the points <italic>p<sub>i</sub> </italic> in U and the functions <italic>f<sub>i</sub> </italic> in U and the properties of the product of automorphic weighted composition operators <inline-formula> <tex-math><?CDATA ${W}_{{f}_{1},{\alpha }_{{p}_{1}}}\,{W}_{{f}_{2},{\alpha }_{{p}_{2}}}\ldots {W}_{{f}_{i},{\alpha }_{pi}}$?></tex-math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mrow> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </msub> <mspace width="0.25em"></mspace> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </msub> <mo>…</mo> <msub> <mi>W</mi> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <mi>p</mi> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> </math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012045_ieqn3.gif" xlink:type="simple"></inline-graphic> </inline-formula> on Hardy space H<sup>2</sup>. In fact, it is very nice connection between analytic function theory and operator theory. In this paper, we give the sufficient and necessary conditions to be normal, unitary, hermitian operator on <italic>H</italic> <sup>2</sup> and we shall present the shape of the numerical range of it.</p>
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Publication Date
Sat Jan 01 2022
Journal Name
1st Samarra International Conference For Pure And Applied Sciences (sicps2021): Sicps2021
Consequences for the groups 𝒮𝓛(2,𝒰), 𝒰 = 23 and 29
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Publication Date
Thu Dec 01 2016
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Synthesis and Characterization of Some Anti-bacterial Active Transition Metal Complexes of N'-[2-(2-Phenyl hydrazinyl) Phenyl] Benzothiohydrazide
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Fri Jan 01 2016
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Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets
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Wed Jun 03 2015
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Baghdad
Synthesis, characterization and Antibacterial of 2, 3 Dihydrobenzo Thiazole -2- Carboxylic acid complexes with some selected Metal ions.
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Wed Jun 18 2014
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Geological Quarterly
Stable carbon and nitrogen isotopes and elemental composition and origin of organic matter from the Euphrates, Injana and Dibdibba formations in Iraq: discrimination between marine and terrestrial environments
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Publication Date
Tue Dec 29 2020
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
Management accounting techniques in product development and achievement customer requirements by adopting the technique of Quality Function Deployment: Applied Research in Baghdad Company for Soft Drinks
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The research aims to apply one of the techniques of management accounting, which is the Quality Function Deployment(QFD) on the Pepsi product in Baghdad Soft Drinks Company and to determine the technical requirements objectively that have been applied in practice in Baghdad Soft Drinks Company / a private shareholding company, as it focuses on meeting the quality requirements and achieving positive quality to provide a product It meets the requirements of current and future customers, hence the importance of research that indicates that the Quality Function Deployment(QFD) is a useful tool to develop the requirements of new products, being a design process driven by customers through their voices, and thus contribute to achieve a competi

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Publication Date
Sun Sep 01 2024
Journal Name
Polish Journal Of Chemical Technology
Preparation of nano SnO<sub>2</sub>-Sb<sub>2</sub>O<sub>3</sub> composite electrode by cathodic deposition for the elimination of phenol by Sonoelectrochemical oxidation
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The preparation of composite metal oxide to attain high efficiency in removing phenol from wastewater has a great concern. In the present study, the focus would be on adopting antimony-tin oxide coating onto graphite substrates instead of titanium; besides the effect of SbCl3 concentration on the SnO2-Sb2O3 composite would be examined. The performance of this composite electrode as the working electrode in the removal of phenol by sonoelectrochemical oxidation will be studied. The antimony-tin dioxide composite electrode was prepared by cathodic deposition with SnCl2 . 2H2O solution in a mixture of HNO3 and NaNO3, with different concentrations of SbCl3. The SnO2-Sb2O3 deposit layer’s structure and morphology were examined and the 4 g/l Sb

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Publication Date
Tue Mar 30 2021
Journal Name
International Journal Of Pharmaceutical Research
Eco-friendly and Efficient Composition, Diagnosis Theoretical, kinetic studies, Antibacterial and Anticancer Activities of Mixed Some Metal Complexes of Tridentate Schiff base Ligand
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Publication Date
Sun Mar 01 2020
Journal Name
International Journal Of Pharmaceutical Research
Eco-friendly and Efficient Composition, Diagnosis Theoretical, kinetic studies, Antibacterial and Anticancer Activities of Mixed Some Metal Complexes of Tridentate Schiff base Ligand
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Abstract: Mixed ligand Mn(II), Co(II), Ni(II), Cu (II), Zn(II), and Cd(II) complexes with (TMAP) Schiff base ligand and (8HQ) have been composition and analyzed. Diagnosis by, melting point, solubility, Electronic, mass and IR-spectroscopic studies, conductivity elemental, thermoanalytical analysis displayed the forming of mononuclear complexes. Spectral studies results suggest an octahedral system or the metal (II) mixed complexes. The detainments of molar conductance of the mixed complexes in DMF coincide to electrolytic nature of the mixed complexes, consequently, these complexes could be subedited as [M(TMAP)(8Q)(H2O)]nX.yH2O (M=Co(II) and Cu(II) complexes(where n = 1, y = 0 ); [M(TMAP)(8Q)(H2O)]nX.yH2O (M = (where n = 1, y = 1 for Ni(

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Publication Date
Tue Dec 30 2025
Journal Name
Acta Scientiarum Polonorum Administratio Locorum
The function of Feng Shui’s five elements in encouraging urban public space for social interaction. Case study: Al-Arabi and Al-Washash neighborhoods in the Al-Mansour district, Baghdad
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Motives: This study examines the integration of Feng Shui principles into urban public spaces to enhance social interaction, livability, and environmental harmony. The case study focuses on the Al-Arabi and Al-Washash neighbourhoods in Baghdad, exploring how spatial characteristics influence engagement and energy circulation (Qi) within the urban network. The motivation arises from deficiencies in connectivity, shading, greenery, and water features, which limit social and aesthetic vibrancy. Aim: The objectives are to evaluate spatial and social dynamics through the Feng Shui Public-Space Index (FS-PSI), Space Syntax analysis, and resident surveys, and to propose strategies using Feng Shui design proxies to improve energy flow, acce

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