In this paper, we find explicit solution to the operator equation $TXS^* -SX^*T^*=A$ in the general setting of the adjointable operators between Hilbert $C^*$-modules, when $T,S$ have closed ranges and $S$ is a self adjoint operator.

We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method dG(q) and the corresponding implicit RungeKutta-Radau method IRK-R(q) of arbitrary order q ≥ 0 for both linear and nonlinear evolution problems of the form u′ + F(u) = f , with γ-angle bounded operator F. The key ingredient is a novel higher order reconstruction Û of the discrete solution U , whi...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

some necessary and sufficient conditions are given for the existence of a g-positive (g-repositive) solution to adjointable operator equations $ax=c,axa^{left( astright) }=c$ and $axb=c$ over hilbert $c^{ast}$-modules, respectively. moreover, the expressions of these general g-positive (g-repositive) solutions are also derived. some of the findings of this paper extend some known results in the...

Consider the homogeneous equation u ′(t) = l(u)(t) for a.e. t ∈ [a, b] where l : C([a, b];R) → L([a, b];R) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvabili...

A propagation method for the time dependent Schrödinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde’s usually results in system of ode’s of the form ut −Gu= s (0.1) where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modifi...

The radiosity equation occurs in computer graphics, and its solution leads to more realistic illumination for the display of surfaces. We consider the behavior of the radiosity integral operator, for smooth and piecewise smooth surfaces. A collocation method for solving the radiosity equation is proposed and analyzed. The method uses piecewise linear interpolation; and for one particular choice...

Given a ®nitely supported sequence a on Z and an s s dilation matrix M, the transition operator Ta is the linear operator de®ned by Tav…a† :ˆ P b2Zs a…Maÿ b†v…b†, where a 2 Z and v lies in `0…Z†, the linear space of all ®nitely supported sequences on Z. In this paper we investigate the spectral properties of the transition operator Ta and apply these properties to the study of the approximation...