# Hrushka Ya. I.

### Base Changeable Sets and Mathematical Simulation of the Evolution of Systems

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1198–1218

We introduce the notion of base changeable sets and study the principal properties of these sets. Base changeable sets are required for the construction of the general theory of changeable sets. The problem studied in our paper is closely connected with the famous sixth Hilbert problem.

### Spaces of generalized operators with bounded projection trace

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 24-39

We construct a theory of Banach spaces of "generalized" operators with bounded projection trace over the given Hilbert space. This theory can be efficient in investigating evolution problems for quantum systems with infinite number of particles.

### Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method

Gorbachuk M. L., Hrushka Ya. I., Torba S. M.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 633–643

For an arbitrary self-adjoint operator *B* in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator *B*, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator *B*, and the *k*-modulus of continuity of the vector *x* with respect to the operator *B*. The results are used for finding *a priori* estimates for the Ritz approximate solutions of operator equations in a Hilbert space.

### On the order of convergence of a semigroup to the identity operator

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 847-851

We describe classes of vectors *f* from a Hilbert space **H** for which the quantity ‖*T*(*t*)*f−f*‖, where *T*(*t*)=*e* ^{−tA }, *t*≥0, and *A* is a self-adjoint nonnegative operator in **H**, has a certain order of convergence to zero as *t*→+0.