# Gorbachuk M. L.

### 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.

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk A. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### 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 $U(t)x$.

### History of the Appearance of Infinite-Dimensional Analysis and its Development in Ukraine

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1058–1073

We present a brief survey of the development of functional analysis in Ukraine and the problems of infinite-dimensional analysis posed and solved for thousands of years, which laid the foundations of this branch of mathematics.

### Yurii Stephanovych Samoilenko (on his 70th birthday)

Berezansky Yu. M., Boichuk A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Nizhnik L. P., Samoilenko A. M., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1408-1409

### Anatolii Mykhailovych Samoilenko (on his 75th birthday)

Berezansky Yu. M., Boichuk A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Perestyuk N. A., Portenko N. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6

### Oleksandr Ivanovych Stepanets’ (on the 70 th anniversary of his birthday)

Gorbachuk M. L., Lukovsky I. O., Makarov V. L., Motornyi V. P., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sharko V. V., Zaderei P. V.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 579-581

### Yuri Yurievich Trokhimchuk (on his 80th birthday)

Berezansky Yu. M., Bojarski B., Gorbachuk M. L., Kopilov A. P., Korolyuk V. S., Lukovsky I. O., Mitropolskiy Yu. A., Portenko N. I., Reshetnyak Yu. G., Samoilenko A. M., Sharko V. V., Shevchuk I. A., Skorokhod A. V., Tamrazov P. M., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 701 – 703

### Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

### Alexander Ivanovich Stepanets

Gorbachuk M. L., Lukovsky I. O., Mitropolskiy Yu. A., Romanyuk A. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1722-1724

### Mark Grigorievich Krein (to the centenary of his birth)

Adamyan V. M., Arov D. Z., Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I., Mikhailets V. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 579-587

### Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 650–657

For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity.

### Evgen Yakovich Khruslov (on his 75 th birthday)

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Pastur L. A., Samoilenko A. M., Sharko V. V.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 549-550

### On the 90th birthday of Yurii Alekseevich Mitropol’skii

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Koshlyakov V. N., Lukovsky I. O., Makarov V. L., Perestyuk N. A., Samoilenko A. M., Samoilenko Yu. I., Sharko V. V., Sharkovsky O. M., Stepanets O. I., Tamrazov P. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 147–151

### Yaroslav Borisovich Lopatinsky (09.11.1906 - 03.10.1981)

Gorbachuk M. L., Lyantse V. É., Markovskii A. I., Mikhailets V. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1443-1445

### 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 behavior of orbits of uniformly stable semigroups at infinity

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 148–159

For uniformly stable bounded analytic $C_0$-semigroups $\{T(t)\} t ≥ 0$ of linear operators in a Banach space $B$, we study the behavior of their orbits $T (t)x, x ∈ B$, at infinity. We also analyze the relationship between the order of approaching the orbit $T (t)x$ to zero as $t → ∞$ and the degree of smoothness of the vector $x$ with respect to the operator $A^{-1}$ inverse to the generator A of the semigroup $\{T(t)\}_{t \geq 0}$. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $e^{-at^{\alpha}}$, where $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ is the angle of analyticity of $\{T(t)\}_{t \geq 0}$, and the collection of these orbits is dense in the set of all orbits.

### Leonid Pavlovych Nyzhnyk (on his 70-th birthday)

Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I., Khruslov E. Ya., Kostyuchenko A. G., Kuzhel' S. A., Marchenko V. O., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1120-1122

### Yurij Makarovich Berezansky (the 80th anniversary of his birth)

Gorbachuk M. L., Gorbachuk V. I., Kondratiev Yu. G., Kostyuchenko A. G., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 3-11

### Functional Analysis in the Institute of Mathematics of the National Academy of Sciences of Ukraine

Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 582–600

We give a brief survey of results on functional analysis obtained at the Institute of Mathematics of the Ukrainian National Academy of Sciences from the day of its foundation.

### 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.

### Dmytro Yakovych Petryna (on his 70 th birthday)

Gorbachuk M. L., Khruslov E. Ya., Lukovsky I. O., Marchenko V. O., Mitropolskiy Yu. A., Pastur L. A., Samoilenko A. M., Skrypnik I. V.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 291-292

### Mykhailo Vasyl'ovych Ostrograds'kyi and his role in the development of mathematics

Gorbachuk M. L., Samoilenko A. M.

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1011-1023

### Yurii Makarovich Berezanskii

Gorbachuk M. L., Mitropolskiy Yu. A., Samoilenko A. M., Skrypnik I. V.

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 579-581

### On analytic solutions of operator differential equations

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 596-607

We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions of a differential equation *y*′(*t*) = *Ay*(*t*), *t* ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya.

### On one generalization of the Berezanskii evolution criterion for the self-adjointness of operators

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 608-615

We describe all weak solutions of a first-order differential equation in a Banach space on (0, ∞) and investigate their behavior in the neighborhood of zero. We use the results obtained to establish necessary and sufficient conditions for the essential maximal dissipativity of a dissipative operator in a Hilbert space.

### Bohdan Iosypovych Ptashnyk

Gorbachuk M. L., Luchka A. Y., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1155–1156

### Behavior of the extended spectral measure of a self-adjoint operator at infinity

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 510–516

We study the dependence of the rate of growth of the extended spectral measure of a self-adjoint operator at infinity on the order of singularity of the vector on which this measure is considered.

### Yurii Dmitrievich Sokolov (on his 100th birthday)

Gorbachuk M. L., Luchka A. Y., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1443-1445

### Estimate of error of an approximated solution by the method of moments of an operator equation

Gorbachuk M. L., Yakymiv R. Ya.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1477-1483

For an equation*Au = f* where*A* is a closed densely defined operator in a Hilbert space*H, f* ε*H*, we estimate the deviation of its approximated solution obtained by the moment method from the exact solution. All presented theorems are of direct and inverse character. The paper refers to direct methods of mathematical physics, the development of which was promoted by Yu. D. Sokolov, the well-known Ukrainian mathematician and mechanic, a great humanitarian and righteous man. We dedicate this paper to his blessed memory.

### Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 55–62

### Inverse problem for a sturm-liouville elliptic equation in a Hilbert space

Ukr. Mat. Zh. - 1991. - 43, № 11. - pp. 1537–1540

### Samuil Davidovich Eidelman (On his sixtieth birthday)

Berezansky Yu. M., Gorbachuk M. L., Ivasyshen S. D., Korolyuk V. S., Mitropolskiy Yu. A., Skrypnik I. V.

Ukr. Mat. Zh. - 1991. - 43, № 5. - pp. 578

### Extension theory for symmetric operators and boundary value problems for differential equations

Gorbachuk M. L., Gorbachuk V. I., Kochubei A. N.

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1299–1313

### Convergence of series of independent Gaussian operators

Gorbachuk M. L., Gorodetskii V. V.

Ukr. Mat. Zh. - 1984. - 36, № 4. - pp. 500 – 502

### Behavior at infinity of solutions of operator differential equations

Gorbachuk M. L., Vynnyshyn Ya. F.

Ukr. Mat. Zh. - 1983. - 35, № 4. - pp. 489—493

### Weak solutions of differential equations in Hilbert space

Gorbachuk M. L., Kashpirovskii A. I.

Ukr. Mat. Zh. - 1981. - 33, № 4. - pp. 513–518

### Scattering problem for first-order differential equations with operator coefficients

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1978. - 30, № 4. - pp. 452–461

### Some questions of the spectral theory of differential equations of elliptic type in the space of vector-functions

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1976. - 28, № 3. - pp. 313–324

### Some questions of the spectral theory of differential equations of elliptic type in the space of vector functions on a finite interval

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1976. - 28, № 1. - pp. 12–26

### The spectrum of self-adjoint extensions of the minimal operator generated by a Sturm-Liouville equation with operator potential

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1972. - 24, № 6. - pp. 726—734

### Classes of boundary-value problems for the sturm — Liouville equation with an operator potential

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1972. - 24, № 3. - pp. 291—304

### Problems of the spectral theory of the second order linear differential equation with unbounded operator coefficients

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1971. - 23, № 1. - pp. 3–14

### On self-adjoint semibounded abstract differential operators

Gorbachuk M. L., Vainerman L. I.

Ukr. Mat. Zh. - 1970. - 22, № 6. - pp. 806—808

### The stability of motion of a solid body containing a liquid-filled cavity suspended by a cord

Gorbachuk M. L., Sleptsova G. P., Temchenko M. Ye.

Ukr. Mat. Zh. - 1968. - 20, № 5. - pp. 586–602

### Basic results of studies at the mathematical-analysis department of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR

Berezansky Yu. M., Gorbachuk M. L.

Ukr. Mat. Zh. - 1967. - 19, № 6. - pp. 73–92

### On the continuation of positively definite functions of two variables

Berezansky Yu. M., Gorbachuk M. L.

Ukr. Mat. Zh. - 1965. - 17, № 5. - pp. 96-102

### On the description of continuations of a positively definite operator function

Ukr. Mat. Zh. - 1965. - 17, № 5. - pp. 102-110

### On the representation of positively determined operator functions

Ukr. Mat. Zh. - 1965. - 17, № 2. - pp. 29-46

### Spectral expansions of Infinitesimal operators of the fourth dimensional group of translations

Ukr. Mat. Zh. - 1964. - 16, № 4. - pp. 528-534