Necessary and sufficient conditions for the operator equation I AXAX n*, to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.

The structural, optical and electrical properties of ZnS films prepared by vacuum
evaporation technique on glass substrate at room temperature and treated at different
annealing temperatures (323, 373, 423)K of thickness (0.5)µm have been studied. The
structure of these films is determined by X-ray diffraction (XRD). The X-ray diffraction
studies show that the structure is polycrystalline with cubic structure, and there are strong
peaks at the direction (111).
The optical properties investigated which include the absorbance and transmittance
spectra, energy band gab, absorption coefficient, and other optical constants. The results
showed that films have direct optical transition. The optical band gab was

CuInSe2(CIS) thin films have been prepared by use vacuum thermal evaporation technique, of thickness750 nm with rate of deposition 1.8±0.1 nm/sec on glass substrate at room temperature and pressure (10-5) mbar. Heat treatment has been carried out in the range (400-600) K for all samples. The optical properties of the CIS thin films are been studied such as (absorption coefficient, refractive index, extinction coefficient, real and imaginary dielectric constant) by determined using Measurement absorption and transmission spectra. Results showed that through the optical constants we can make to control it are wide applications as an optoelectronic devices and photovoltaic applications.

 X-ray diffraction pattern reveled the tetragonal crystal system of SnO2  Thin films of SnO2 were prepared on glass substrates using Spray Pyrolysis Technique. The absorption and transmition spectra were recorded in the rang of 300-900nm,  the   spectral   dependences   of   absorption coefficient were calculated   from   transmission spectra. The direct and allowed optical  energy gap has been evaluated from plots of   (αhÏ…)²  vs. (hÏ…) . The energy gap was found to be  2.4-2.6eV.  The optical constant such as extinction coefficient( k )  and absorption coefficient ( α) have been evaluated. 

In this research thin films of (CdTe) have been prepared as pure and doped by Zn
with different ratios (1,2,3,4,5)% at thickness (400+25)nm with deposition rate (2±0.1)nm,
deposited on glass substrate at R.T. by using thermal evaporation in vacuum . All samples
were annealed at temperature (523,573,623,673)K at 1h.
The structural prop erties of all prepared thin films, doped and undoped have been
studied by using XRD. The analysis reveals that the structures of the films were
polycrystalline and typed cubic with a preferred orientation along (111) plane for the
undoped films with (2,3)% of zinc , and shifting (2ÆŸ) for doped films . The annealing films
at temperature 573 K and Zn:3% show decreasing in

Zinc Oxide (ZnO) thin films of different thickness were prepared
on ultrasonically cleaned corning glass substrate, by pulsed laser
deposition technique (PLD) at room temperature. Since most
application of ZnO thin film are certainly related to its optical
properties, so the optical properties of ZnO thin film in the
wavelength range (300-1100) nm were studied, it was observed that
all ZnO films have high transmittance (˃ 80 %) in the wavelength
region (400-1100) nm and it increase as the film thickness increase,
using the optical transmittance to calculate optical energy gap (Eg
opt)
show that (Eg
opt) of a direct allowed transition and its value nearly
constant (~ 3.2 eV) for all film thickness (150

Compounds from ZnO doped with AgO in different ratio (0,3,5,7, and 9)wt.% were prepared.Thin films from the prepared compounds were deposited on a glass substrate using the pulsed laser deposition method. The XRD pattern confirmed the presence of a single-phase hexagonal wurtzite ZnO structure, without the presence of a secondary phase. AFM measurements showed an increase in both average grain size and average surface roughness with increasing concentration content of (AgO).The crystallite size of ZnO of the main peak corresponding to the preferred plane of crystal growth named (100) increases from 17.8 to 22.5nm by increasing of AgO doping ratio  from 0 to 9%. The absorbance and transmittance in the wavelength range (350-1100 nm) were

Thin films of  BhSe3  have being deposited on glass substrates of

about 80 - 172 ± 14 nm thickness from an aqueous solution bath at temperature 293 K for period 0.5 to 6.0 hours  using alchemical bath deposition method .

The  films  are  characterized   by  X-ray  diffraction,     X-ray

florescent techniques and optical transmittance spectra measurements in the rang 350 - 400 nm at 293 K. And shows that as deposited  films are amorphous and a  transition to polycrystalline state has taken place after  annealing  them  at  373  K,  for  30  minutes,  But  they  will  be dan1aged