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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 Apr 01 2023
Journal Name
Baghdad Science Journal
Physiological and Hormonal Effects of Titanium Dioxide Nanoparticles on Thyroid Function and the Impact on Bodyweight in Male Rats
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Fifty-four Sprague-Dawley albino adult male rats were classified into three main groups each of 18 rats treated for a particular duration (1,2, and 4) weeks respectively. Each group was subdivided into three subgroups each of six rats treated as follows; group (1) serve as normal control, group (2, and 3) intra-peritoneal treated with TiO2NPs (50,200) mg/kg respectively, body *weight of all rats was measured before and after the experiment, then rats were dissected at the end of each experiment and the weights of the thyroid was measured. The result showed a highly significant decrease (p<0.01) in thyroid gland weight, a highly significant increase (p<0.01) in body weights and TSH, while a highly significant decrease (p&

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Publication Date
Mon Jan 01 2018
Journal Name
Journal Of Engineering And Applied Sciences
sEffect of Sb doping on CuAlSe2 thin films and their behavior on the preparation CuAlSe2/Si heterojunction solar cells
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Publication Date
Sun Jan 01 2023
Journal Name
International Journal Of Nonlinear Analysis And Applications
Rock facies classification and its effect on the estimation of original oil in place based on petrophysical properties data
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The most significant function in oil exploration is determining the reservoir facies, which are based mostly on the primary features of rocks. Porosity, water saturation, and shale volume as well as sonic log and Bulk density are the types of input data utilized in Interactive Petrophysics software to compute rock facies. These data are used to create 15 clusters and four groups of rock facies. Furthermore, the accurate matching between core and well-log data is established by the neural network technique. In the current study, to evaluate the applicability of the cluster analysis approach, the result of rock facies from 29 wells derived from cluster analysis were utilized to redistribute the petrophysical properties for six units of Mishri

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Publication Date
Sat Oct 30 2021
Journal Name
Iraqi Journal Of Science
The Effects of Conductance on Metastable Switches in Memristive Devices Based on Anti-Hebbian and Hebbian (AHaH) Learning Rules
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     In the last few years, the literature conferred a great interest in studying the feasibility of using memristive devices for computing. Memristive devices are important in structure, dynamics, as well as functionalities of artificial neural networks (ANNs) because of their resemblance to biological learning in synapses and neurons regarding switching characteristics of their resistance. Memristive architecture consists of a number of metastable switches (MSSs). Although the literature covered a variety of memristive applications for general purpose computations, the effect of low or high conductance of each MSS was unclear. This paper focuses on finding a potential criterion to calculate the conductance of each MMS rather t

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Publication Date
Wed Jan 01 2020
Journal Name
Plant Archives
Morpho-physiological study on the effect of lead stress and selenium foliar application on growth of dill plant anethum graveolens l
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The experiment was carried out in the green house of botanical garden belong to Department of Biology/College of Education for Pure Science Ibn AL-Haitham, University of Baghdad for growing season 2017-2018 to evaluate effect of lead stress with concentrations (0, 50, 100, 150) mg.L -1 and Selenium concentrations (0, 15, 30) mg.L-1 on growth of dill plant using pots. The experiment was designed according to completely randomized design (CRD) with three replications. Result indicated that dill plants subjected to lead stress with height concentrations caused decrease in plant parameters (plant height, no. of branches. plant-1, root length, shoot dry weight, the content of nitrogen, phosphorus and potassium, protein concentration, no. of umbe

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Publication Date
Mon Oct 01 2012
Journal Name
Iraqi Journal Of Physics
Study the effect of adding TiO2 nano-powder on some surface properties of TiO2 thin film prepared on stainless steel substrate
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In this research TiO2 nano-powder was prepared by a spray pyrolysis technique and then adds to the TiO2 powder with particle size (0.523 μm) in ratio (0, 5, 10, 15 at %) atomic percentage, and then deposition of the mixture on the stainless steel 316 L substrate in order to use in medical and industrial applications.
Structure properties including x-ray diffraction (XRD) and scanning electron microscope (SEM0, also some of mechanical properties and the effect of thermal annealing in different temperature have been studied. The results show that the particle size of a prepared nano-powder was 50 up to 75 nm from SEM, and the crystal structure of the powders (original and nano powder) was rutile with tetragonal cell. An improvement in

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Publication Date
Sat Jun 01 2019
Journal Name
Journal Of Economics And Administrative Sciences
Analysis of the relationship between Government spending compression and the window of the currency and its impact on the exchange rate of the Iraqi dinar
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The exchange rate is the backbone of  any economy in the world, whether  developed or developing, where most countries adopted  many policies, in order to ensure the stability of the exchange rate of the currency, because of its importance as a link between the local economy and the others ,And it contribute in the achievement of internal and external balance and despite the many different factors that affect it, but there is wide consensus on the effectiveness of the role of spending and the currency window in the exchange rate of the Iraqi dinar, especially in the Iraqi economy, effectiveness As the increase in government spending lead to an increase in the supply of money and increase domestic demand and high pr

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Publication Date
Thu Sep 01 2011
Journal Name
Journal Of Economics And Administrative Sciences
The impact of political thinking on the decision maker in analyzing the strategic environment of the state (applied study in the Iraqi Ministry of Finance)
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The Political Thinking Regarded as an important element for the formulation of the stat, weather in its formation, the structure of it s entity, its political system and it s governmental instruments .The political thinking can not act without determined strategy, So they intend to work hard to formulate a railed strategy that make them able to determine its  directions to general issues.

   The Study aimed to solve the problem through the following question:

1- What are the levels of Political Thinking and Strategic Analysis in the financial ministry?

2- What are the relation ship between the dimensions of Political  T

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Publication Date
Fri Mar 28 2025
Journal Name
Journal Of Physical Education
Special exercises using tools and their effect on teaching the skill of a front shoulder circle on the rings apparatus in artistic gymnastics for students
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Sports skills in some individual games require physical and motor qualities to facilitate the process of skill performance and also require the instructor or trainer to use more than one strategy, method and way to bring the performance to the level of mastery and avoid injury. The aim of the research is to know the effect of using special exercises using tools and their effect on teaching the skill of a front shoulder circle. The research hypothesis is that using special exercises with tools has a positive effect on teaching the skill of a front shoulder circle on the rings apparatus. Research method: - The researchers used the experimental method by designing two equal groups, the control and the experimental, to suit the research

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Publication Date
Fri Sep 01 2023
Journal Name
Journal Of Molecular Structure
Synthesis, spectral studies, DFT, biological evaluation, molecular docking and dyeing performance of 1-(4-((2-amino-5-methoxy)diazenyl)phenyl) ethanone complexes with some metallic ions
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