A posteriori choice of time-discretization step in finite difference methods for solving ill-posed Cauchy problems in Hilbert space

Finite difference semidiscretization methods for solving an ill-posed Cauchy problem in a Hilbert space are investigated. The problems involve linear positively definite selfadjoint operators. We justify an a posteriori scheme for the choice of the time-discretization step and establish accuracy estimates in terms of the error level of input data.

On One Method of Studying Spectral Properties of Non-selfadjoint Operators

In this paper, we explore a certain class of Non-selfadjoint operators acting on a complex separable Hilbert space. We consider a perturbation of a nonselfadjoint operator by an operator that is also nonselfadjoint. Our consideration is based on known spectral properties of the real component of a nonselfadjoint compact operator. Using a technique of the sesquilinear forms theory, we establish the compactness property of the resolvent and obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of nonselfadjoint operators. We obtain a classification of nonselfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vector system. Finally, we obtain an asymptotic formula for the eigenvalues.

Some integral inequalities for operator monotonic functions on Hilbert spaces

Let f be an operator monotonic function on I and A, B∈I (H), the class of all selfadjoint operators with spectra in I. Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I,\matrix{0 \hfill & { \le \int\limits_0^1 {p\left( t \right)f\left( {\left( {1 - t} \right)A + tB} \right)dt - \int\limits_0^1 {p\left( t \right)dt\int\limits_0^1 {f\left( {\left( {1 - t} \right)A + tB} \right)dt} } } } \hfill \cr {} \hfill & { \le {1 \over 4}\left[ {p\left( 1 \right) - p\left( 0 \right)} \right]\left[ {f\left( B \right) - f\left( A \right)} \right]} \hfill \cr }in the operator order.Several other similar inequalities for either p or f is differentiable, are also provided. Applications for power function and logarithm are given as well.