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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
Sun Mar 26 2017
Journal Name
Iraqi Journal Of Pharmaceutical Sciences ( P-issn 1683 - 3597 E-issn 2521 - 3512)
Estimation of Oxidative Stress and Some Trace Elements in Iraqi Men Patients with Type 2 Diabetes Mellitus
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Type 2 diabetes mellitus (T2DM) is a chronic disorder that is associated with the imbalance of trace elements which are involved in many functions especially enzyme activities. Changes in the levels of serum elements probably can create some complications in type 2 diabetes mellitus.  Previous experimental and clinical studies report that oxidative stress plays a major role in the pathogenesis and development of (T2DM). However, the exact mechanism of oxidative stress could contribute to and accelerate the development of (T2DM).

The aim of this study contained the following sections: firstly, to determine some biochemical parameters in subjects with type 2 diabetes mellitus (T2DM) like lipid peroxidation marker, malondialdeh

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Publication Date
Sun Jul 01 2018
Journal Name
Journal Of Global Pharma Technology
Synthesis of New 2, 4, 5-triphenyl imidazole Derivatives Derived from benzoin and Studying their Biological Activity
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compound [1] was formed from the reaction of benzoin and benzaldehyde in the presence of ammonia, which was reacted with sodium hydride in DMF to obtain imidazole salt. This salt was reacted with adipoyl chloride to give compound [2]. Acid hydrazide derivative [3] was obtained from the reaction of compound [2] with hydrazine hydrate. After that Shiff bases [4-9] have been synthesized from the reaction of compound [3] with different aromatic aldehydes. These new formed compounds were diagnosed by 13C-NMR, 1H-NMR for some of them (in Ahl-Albate University in Jordan) and FT-IR spectroscopy (In Baghdad University). All of the prepared products have been studied their biological activities toward two kinds of bacteria. These products show

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Publication Date
Sun Jun 05 2016
Journal Name
Baghdad Science Journal
Evaluation of Serum Malondialdehyde, Glutathione and Lipid Profile Levels in Iraqi Females with Type 2 Diabetes Mellitus
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This study is carried out on patients with type 2 diabetes mellitus to assess the lipid profile, malondialdehyde and glutathione. Our study is concerned with 51 (Iraqi Arab females) patients of type 2 diabetes mellitus compared with 31 control subjects unified in age, sex and ethnic background. Lipid profile is measured by using commercially available kits, while the serum MDA and glutathione levels are measured by means of sandwich ELISA test using commercially available kits. Serum MDA is significantly higher (P<0.001) while glutathione is significantly lower (P<0.001) in type 2 diabetic patients when compared to the control. The normal levels of MDA (3.82 ± 0.77n mol/ml) and GSH (2.23 ± 0.54 µg/ml) recorded for the non-diabetic female

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Publication Date
Thu Aug 31 2017
Journal Name
Journal Of Engineering
Study and Analysis of Concentric Shell and Double Tube Heat Exchanger Using - Al 2 O 3 Nanofluid
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    Heat exchanger is an important device in the industry for cooling or heating process. To increase the efficiency of heat exchanger, nanofluids are used to enhance the convective heat . transfer relative to the base fluid. - Al2O3/water nanofluid is used as cold stream in the shell and double concentric tube heat exchanger counter current to the hot stream basis oil. These nanoparticles were of particle size of 40 nm and it was mixed with a base fluid (water) at volume
concentrations of 0.002% and 0.004%. The results showed that each of Nusselt number and overall heat transfer coefficient increased as nanofluid concentrations increased. The pressure drop of nanofluid increased slightly than the base fluid because

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Publication Date
Thu Dec 01 2022
Journal Name
Baghdad Science Journal
Synthesis, Spectral Identification, Antibacterial Evaluationand Theoretical Study of Co, Fe, Rh and Pd Complexes for 2-benzoylthiobenzimidazol
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A new novel series of metalcomplexes are prepared from reactions between 2-benzoylthio- benzimidazole (L) with metal salts of Co (II) , Fe(III) and Rh (III) , while Pd(II) complex was obtained by mixing ligandsof 2-benzoylthiobenzimidazole (L) as primary ligand and bipyridine (L/)as secondary ligand as well as palladium chloride as metal salt in an ethanoic medium. The geometry of these compounds were identified using C.H.N.microanalysis, Ultraviolet–visible, Fourier transforms infrared, magnetic susceptibility, molar conductivity and flame atomic absorption (A.A). From the dataobtained by these spectral analyses, the molecular structures for Rh and Fe complexes were proposed to be octahedral geometry. A square planar const

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Publication Date
Sat Sep 11 2010
Journal Name
Journal Of Al-nahrain University
ESTIMATION ACTIVITY OF LAP IN PATIENT S WITH TYPE 2 DIABETES BY USING LEUCINE AMIDE AS SUBSTRATE
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This study was performd on 50 serum specimens of patients with type 2 diabetes, in addition, 50 normal specimens were investigated as control group. The activity rate of LAP in patients (560.46 10.504) I.U/L and activity rate of LAP in healthy(10.58 4.39)I.U/L.The results of the study reveal that Leucine aminopeptidase (LAP) activity of type 2 diabetes patient s serum shows a high signifiacant increase (p < 0.001) compare to healthy subjects. Addition preparation leucine amide as substrate of LAP, identification melting point and spectra by FTIR. K

Publication Date
Sun Jun 05 2016
Journal Name
Baghdad Science Journal
Complexes of Some Transition Metal with 2-Benzoyl thiobenzimidazole and 1,10-Phenanthroline and Studying their Antibacterial Activity
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Mixed ligands of 2-benzoyl Thiobenzimiazole (L1) with 1,10-phenanthroline (L2) complexes of Cr(III) , Ni(II) and Cu(II) ions were prepared. The ligand and the complexes were isolated and characterized in solid state by using FT-IR, UV-Vis spectroscopy, 1H, 13C-NMR, flame atomic absorption, elemental micro analysis C.H.N.S, magnetic susceptibility , melting points and conductivity measurements. 2-Benzoyl thiobenzimiazole behaves as bidenetate through oxygen atom of carbonyl group and nitrogen atom of imine group. From the analyses Octahedral geometry was suggested for all prepared complexes. A theoretical treatment of ligands and their metal complexes in gas phase were studied using HyperChem-8 program, moreover, ligands in gas phase

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Publication Date
Fri Mar 31 2017
Journal Name
Journal Of Engineering
Cultivation of Chlorella Vulgaris Using Airlift Photobioreactor Sparged with 5%CO 2 -Air as a Biofixing Process
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The present paper addresses cultivation of Chlorella vulgaris microalgae using airlift photobioreactor that sparged with 5% CO 2 /air. The experimental data were compared with that obtained from bioreactor aerated with air and unsparged bioreactor. The results showed that the concentration of biomass is 0.36 g l -1  in sparged bioreactor with CO2/air, while, the concentration of biomass reached to 0.069 g l -1  in the unsparged bioreactor. They showed also that aerated  ioreactor.with CO2/air gives more biomass production even the bioreactor was aerated with air. This study proved that application of sparging system for  ultivation of Chlorella vulgaris microalgae using either CO2/air mixture or air has a significant

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Publication Date
Thu Dec 01 2022
Journal Name
Baghdad Science Journal
Synthesis, Spectral Identification, Antibacterial Evaluationand Theoretical Study of Co, Fe, Rh and Pd Complexes for 2-benzoylthiobenzimidazol
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A new novel series of metalcomplexes are prepared from reactions between 2-benzoylthiobenzimidazole (L) with metal salts of Co (II) , Fe(III) and Rh (III) , while Pd(II) complex was obtained by mixing ligandsof 2-benzoylthiobenzimidazole (L) as primary ligand and bipyridine (L/ )as secondary ligand as well as palladium chloride as metal salt in an ethanoic medium. The geometry of these compounds were identified using C.H.N.microanalysis, Ultraviolet–visible, Fourier transforms infrared, magnetic susceptibility, molar conductivity and flame atomic absorption (A.A). From the dataobtained by these spectral analyses, the molecular structures for Rh and Fe complexes were proposed to be octahedral geometry. A square planar construction is propo

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Publication Date
Sun Jun 04 2023
Journal Name
Iraqi Journal Of Biotechnology,
Gene Expression of Adenosine Deaminase Genes 1 and 2 in Female Iraqi Patients with Autoimmune Thyroid Disease
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The current study was carried out to investigate the correlation of gene expressions of ADA1 and ADA2 genes with the development of autoimmune thyroid disease (AITD) in a sample of Iraqi females. One hundred patients with AITD and 80 controls were included. Quantitative real time polymerase chain reaction (qRT–PCR) was utilized for investigation of ADA1 and ADA2 gene expression among patients and controls. The correlation of age and body mass index (BMI) with AITD occurrence comparing with controls was studied. Based on the results of this study, there is high expression level of ADA1 and ADA2 genes in patients compared with healthy controls; also, the gene expression fold (2-ΔΔCT) of ADA1 and ADA2 among AITD patients was recorded and a

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