# Gorbachuk V. M.

### Spaces of smooth and generalized vectors of the generator of an analytic semigroup and their applications

Gorbachuk M. L., Gorbachuk V. M.

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 478-509

For a strongly continuous analytic semigroup $\{ e^{tA}\}_{t\geq 0}$ of linear operators in a Banach space $B$ we investigate some locally convex spaces of smooth and generalized vectors of its generator $A$, as well as the extensions and restrictions of this semigroup to these spaces. We extend Lagrange’s result on the representation of a translation group in the form of exponential series to the case of these semigroups and solve the Hille problem on description of the set of all vectors $x \in B$ for which there exists $$\mathrm{l}\mathrm{i}\mathrm{m}_{n\rightarrow \infty }\biggl( I + \frac{tA}n \biggr)^n x$$ and this limit coincides with etAx. Moreover, we present a short survey of particular problems whose solutions are necessary for the introduction of the above-mentioned spaces, namely, the description of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator; the representation of solutions to operator-differential equations on an open interval and the analysis of their boundary values, and the existence of solutions to an abstract Cauchy problem in various classes of analytic vector-valued functions.

### Representations of a Group of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

Gorbachuk M. L., Gorbachuk V. M.

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 592-601

For a strongly continuous one-parameter group $\{U(t)\} t ∈(−∞,∞)$ of linear operators in a Banach space $\mathfrak{B}$ with generator $A$, we prove the existence of a set $\mathfrak{B}_1$ dense in $\mathfrak{B}_1$ on the elements $x$ of which the function $U(t)x$ admits an extension to an entire B$\mathfrak{B}$-valued vector function. The description of the vectors from $\mathfrak{B}_1$ for which this extension has a finite order of growth and a finite type is presented. It is also established that the inclusion $x ∈ \mathfrak{B}_1$ is a necessary and sufficient condition for the existence of the limit ${ \lim}_{n\to 1}{\left(I+\frac{tA}{n}\right)}^nx$ and this limit is equal to $U(t)x$.

### On the correct solvability of the Dirichlet problem for operator differential equations in a Banach space

Gorbachuk M. L., Gorbachuk V. M.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1462–1476

We investigate the structure of solutions of an equation $y″(t) = By(t)$, where $B$ is a weakly positive operator in a Banach space B, on the interval $(0, \infty)$ and establish the existence of their limit values as $t → 0$ in a broader locally convex space containing $B$ as a dense set. The analyticity of these solutions on $(0, \infty)$ is proved and their behavior at infinity is studied. We give conditions for the correct solvability of the Dirichlet problem for this equation and substantiate the applicability of power series to the determination of its approximate solutions.

### On the uniqueness of solutions of the Dirichlet and Neumann problems for an elliptic second-order differential equation on a semiaxis

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1564–1567

For an elliptic second-order differential equation in a Banach space, we give a description of all solutions of the homogeneous Dirichlet and Neumann problems and establish conditions under which these problems are uniquely solvable.

### Behavior at infinity of solutions of an operator differential equation of first order in a Banach space

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 629-631