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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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Comparative study of oral bacterial composition and neutrophil count between smokers and non-smokers
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Enhanced optical response and photodetector performance of PANi/WO3 nanocomposites with controlled composition ratios
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In recent years, polymeric nanomaterials have attracted much attention to the fields of optoelectronics and energy storage due to their unique properties. In this study, to improve limitations, such as moderate conductivity and limited stability of polyaniline (PANi), PANi /WO3 nanocomposites with different amounts of tungsten trioxide nanoparticles (2,5, 5 and 10 wt%) were synthesized and investigated. The samples were prepared using the rotation coating method and evaluated in structural tests (XRD, FTIR, FESEM), optical (UV-Visible, PL) and photosensitivity. The results showed that the presence of WO3 caused a gradual decrease in the energy band gap from 3.04 eV in pure PANi to 2.37 eV in the 10 wt% sample, which led to facilitating elec

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Mesoporous activated carbon from eggshell waste via H₃PO₄ chemical activation and its application in the removal of Janus Green B dye
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The main objective of this study was to evaluate the adsorption efficiency of two adsorbent materials, Iraqi chicken eggshells (ESh) and activated carbon (AC) derived from ESh powder for the removal of a cationic dye (Janus green B; JGD) from aqueous solution. Activated carbon was synthesised from ESh using a simple chemical activation method using phosphoric acid as the activating agent. The physicochemical properties of the adsorbents were characterised by the Brunauer–Emmett–Teller (BET) method, FT-IR spectroscopy, scanning electron microscopy (SEM), energy-dispersive X-ray spectroscopy (EDS), inductively coupled plasma optical emission spectroscopy (ICP-OES), and point of zero charge (pHpzc). The results of BET analysis confirmed th

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Mesoporous activated carbon from eggshell waste via H₃PO₄ chemical activation and its application in the removal of Janus Green B dye
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The main objective of this study was to evaluate the adsorptionefficiency of two adsorbent materials, Iraqi chicken eggshells (ESh)and activated carbon (AC) derived from ESh powder for theremoval of a cationic dye (Janus green B; JGD) from aqueoussolution. Activated carbon was synthesised from ESh usinga simple chemical activation method using phosphoric acid asthe activating agent. The physicochemical properties of the adsor-bents were characterised by the Brunauer–Emmett–Teller (BET)method, FT-IR spectroscopy, scanning electron microscopy(SEM), energy-dispersive X-ray spectroscopy (EDS), inductivelycoupled plasma optical emission spectroscopy (ICP-OES), andpoint of zero charge (pHpzc). The results of BET analysis confirmedthat AC has

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Determine The Factors Affecting Product Quality Using FMEA Analysis: A Case Study in Al Zawraa State Company
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Applying the substance-field model mechanism to problem solving in industrial product design: محمد علي حسين القيسي
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