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 $TXS^*-SX^*T^*= A$.

L = ∆g + aI Here a ∈ L∞(M) and ∆gu = −divg(∇u) (Or for simplicity you can choose a ∈ C∞). And we define the kernel Ka (depending on the constant a) of the differential operator L: Ka = KerL ∩W 1,2 0 (M) It is obvious that if the differential operator is coersive, then the Kernel Ka = 0. Definition 1. (Green Function) Define G : M̄×M̄\diag{M̄} → R is a Green function for L with Dirichelet Boundary ...

In this paper, the maximal dissipative extensions of a symmetric singular 1D discrete Hamiltonian operator with maximal deficiency indices (2,2) (in limit-circle cases at ±∞) and acting in the Hilbert space ℓ_{Ω}²(Z;C²) (Z:={0,±1,±2,...}) are considered. We consider two classes dissipative operators with separated boundary conditions both at -∞ and ∞. For each of these cases we establish a self...