In this paper, we state some results on product of operators with closed ranges and we solve the operator equation $TXS^*-SX^*T^*= A$ in the general setting of the adjointable operators between Hilbert $C^*$-modules, when $TS = 1$. Furthermore, by using some block operator matrix techniques, we nd explicit solution of the operator equation $TXS^*-SX^*T^*= A$.

We have used a variational approach to calculate some thermodynamic properties of the quasi-one dimensional liquid 3He such as the energy, entropy, free energy, equation of state and heat capacity at finite temperature. We have employed the Lennard-Jones potential as the inter-atomic interaction. We have seen that the total energy increases by increasing both temperature and density....

A semi-empirical equation of state has been studied for modelling vapour-liquid data of pure substances. The specific molecular based equation of state is employed here as basis because of its mathematical simplicity. The semi-empirical extension has been accomplished to real fluids by correlating the density dependence of the attraction term to vapour liquid data of a reference fluid. The resu...