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  {{\bf Marko Grygorovych Krein} \\
    (to the 100th birthday anniversary)}
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April 3, 2007, is the day of 100-th birthday anniversary of Marko
Grygorovych Krein, one of the most celebrated mathematicians of the
20-th century, whose whole life was closely connected with Ukraine.
He was born in Ky\"\i{}v in a family, with a modest income, that had
seven children. His father had a timber selling business, and, after
the 1917 Revolution, had to leave this business.

Marko Grygorovych has exhibited his extraordinary mathematical
talents during his teens.  Beginning in fourteen, he started to
systematically attend lectures of D.~O.~Grave and scientific
seminars of D.~O.~Grave and B.~M.~Delone in Ky\"\i{}v University,
attended lectures of B.~M.~Delone in Ky\"\i{}v Polytechnic
Institute. At the age of 17, influenced by the work of M.~Gorky ``My
Universities'', he decided that it is the time to start his
``universities'' and, together with his friend, he went to Odessa to
join one of circus troupes, for he had a dream of becoming an
acrobat.  However, the fate had its way, and saved to the world, in
the person of M.~G.~Krein, not an acrobat but a prominent young
mathematician whose influence on the development of mathematics can
not be overestimated.  The acrobat's job has been already occupied.
Waiting for a new opening, he met N.~G.~Chebotarev, a famous
algebraist and a wonderful person, to whom he had a recommendation
letter from D.~O.~Grave. At that time, M.~G.~Chebotarev conducted a
research in Odessa University. Feeling the mathematical gifts of the
young person, he convinced him not to follow his circus dreams and
have prepared him for a PhD school. Together with S.~I.~Shatunovski,
they succeeded in getting a permission for admitting M.~G.~Krein,
who was 19, to a Doctorate Program, although Krein did not have even
a high school diploma, to say nothing about a university degree. In
this way, he became a PhD student in Odessa University with
M.~G.~Chebotarev guidance. Since then, M.~G.~Krein always had a
photograph of M.~G.~Chebotarev over his desk. In his ``Mathematical
Autobiography'', N.~G.~Chebotarev, talking about 17 year old
M.~G.Krein, recalled that he ``without having graduated from a high
school, had brought a personal work with a very distinguished
content''. N.~G.~Chebotarev was very proud of his first student and
regarded him as ``one of the best mathematicians in Ukraine''.

A well-known specialist in Mechanics G.~K.~Suslov got also
interested in working with M.~G.~Krein. Together with
F.~R.~Gantmacher, M.~G.~Krein were attending his seminar in Odessa
Polytechnic Institute. Later interests of Mark Krein were formed
under a direct influence of M.~~G.~Chebotarev and G.~K.~Suslov.
M.~G.~Chebotarev gave him love for algebraic techniques and for
algebra in general, an interest in various problems in the theory of
functions, in particular, the problem of distributions of zeros for
some classes of functions, interpolation and extension theory, and
an interest in Mechanics came from working with G.~K.~Suslov; this
interest can be traced in many mathematical works of M.~G.~Krein.

In 1928, M.~G.~Chebotarev had moved to Kazan' and became a Professor
of Kazan' University. A year after that, M.~G.~Krein have finished
his Doctoral studies and started to teach in Donetsk Mining
Institute, where he have been working for two years. At the time, he
was already married; in 1927 he have happily married Raisa L'vovna
Romen, a faithful friend and an assistant for sixty years. She was
specializing in the ship architecture, having graduated from Odessa
Marine Institute.  Their only child, daughter Irma, have obtained a
PhD degree in Philosophy, became a specialist in Cybernetics, and
founded a new branch the ``Humanitarian Cybernetics''. The only
grandson Alexey, who having graduated from the Department of
Mathematics in Odessa University and worked in the theory of
systems, has died at a very early age from a blood illness, which
has greatly influenced the health of Krein and his wife, with whom
he lived all his short life.  The only grand grandson they have,
also a mathematician, now lives outside of Ukraine.

In 1931, M.~G.~Krein has come back to Odessa, and took the position
of a Professor at Odessa University. He worked together with
B.~Ya.~Levin, with whom he was in friendly relations starting with
their first meeting and until his last days. In 1934, M.~G.~Krein
has obtained a Doctor degree in Physics and Mathematics from Moscow
University without submitting a thesis at the age of 31, and shortly
after that, in 1939, he was elected a corresponding member of the
Academy of Sciences of the Ukrainian Soviet Socialist Republic.

The early flourishing of M.~G.~Krein's talent as a scientist was
accompanied by as early opening of his pedagogical talents. At the
age of 25, he has started, in Odessa University, a scientific
seminar that soon became one of the leading centers for the research
in functional analysis, a young branch of mathematics at the time
that became the main area of his research. During this period,
M.~G.~Krein's mathematical interests include oscillation matrices
and kernels, geometry of Banach spaces, the Nevanlinna--Pick
interpolation problem, extension of positive definite functions and
their applications. His first students were A.~V.~Artemenko,
M.~S.~Lifshets, D.~P.~Mil'man, M.~A.~Naimark, M.~A.~Rutman,
S.~A.~Orlov, V.~L.~Shmul'yan. Works of these mathematicians became
an indispensable part of the modern mathematics.

At the same time, M.~G.~Krein also worked in the Research Institute
at Khar'kov University (1934 - 1940), and during the periods
1940~--~1941 and 1944~--~1952, he was the Head of the Department of
Algebra and Functional Analysis at the Institute of Mathematics of
the Academy of Sciences of the Ukrainian SSR (one of the researchers
working here during the period of 1940--1941 was great S.~Banach
with whom M.~G.~Krein had scientific contacts since his trip to
L'vov in 1940). Many of his results obtained at that time, as well
as results obtained with his students, friends and colleagues,
including N.~I.~Akhiezer and F.~R.~Gantmacher, now became classic
and can be found in major monographs and textbooks on functional
analysis.

During the World War~II, M.~G.~Krein headed the Chair of Theoretical
Mechanics at Kuibyshev Industrial Institute (Russia). He had
preferred working there as opposed to the Chair of Mathematics,
since for a technical school, it deals with more scientific
directions and gives wider possibilities.

In 1944, he returned back to Odessa and never left it again. He
loved this city, new history of its streets, was fond of the local
``Odessa language'', Odessa jokes; he often visited Odessa Operetta.
However, very soon M.~G.~Krein was laid off the Odessa University.
His closest friend B.~Ya.~Levin could not also work there any more.
This was a consequence of the antisemitic politics conducted by
Stalin regime and the corruption of the University administration.
A principal scientific attitude of these scientists, their
opposition to pushing through illiterate doctorate theses, was
regarded as an indication of Zionism. The official notice about his
laying off, M.~G.~Krein has obtained in the day of his forty years
old anniversary as a ``present'' from his directors that favored
``more reliable'' staff, taking into account the political situation
in the country and the ``new personal policy'' in forties, which was
conducted under the thesis of fighting Zionism and Cosmopolitanism.
This meant an end of the center of functional analysis in Odessa
University, an end of the official scientific carrier of
M.~G.~Krein.

In 1944~--~1954, M.~G.~Krein was working in the Department of
Theoretical Mechanics in the Odessa Marine Institute. Regardless the
difficulties pertaining these times, he has founded a number of
important directions in mathematics and mechanics, became a world
famous scientist. Together with the theoretical value of his
results, their applied importance has also increased, especially
those related to parametric resonance. V.~Veksler, a renown
physicist, has remarked that ``without works of M.~G.~Krein, we
would not have a synchrophasotron''. In a popular book of N.~Wiener,
a father of cybernetics, ``I Am a Mathematician'', the name of
M.~G.~Krein goes together with the name of A.~M.~Kolmogorov, which
was a way to acknowledge the value of their researches, published in
the Proceedings of the Academy of Sciences of the USSR, in the
estimation and control theories during and shortly after the war.
New official and unofficial students of M.~G.~Krein are renown
mathematicians and physicists I.~Ts.~Gohberg, I.~S.~Iokhvidov,
I.~S.~Kats, A.~A.~Kostyukov, G.~Ya.~Lyubarskii, A.~A.~Nudel'man,
G.~Ya.~Popov, V.~G.~Sizov, Yu.~L.~Shmul'yan.

In 1952, M.~G.~Krein was laid off the Institute of Mathematics of
the Academy of Sciences of the Ukrainian SSR, where he has also
founded a school of functional analysis, representatives of which
are Yu.~M.~Berezansky, Yu.~L.~Daletsky, G.~I.~Kats,
M.~O.~Krasnosel'skii, B.~I.~Korenblyum, S.~G.~Krein. The official
explanation was that he lived in Odessa but not in Kiev. However, as
it recalls I.~Ts.~Gohberg in his memoirs, ``it is easy to twig: at
this time there was a known tragedy with Jewish doctors''.

Starting in 1954 and until the retirement, Marko Grygorovych had the
Chair of Theoretical Mechanics in Odessa Institute of Civil
Engineering. At the end of his life, he worked as a consultant in
the Institute of Physical Chemistry of the Academy of Sciences of
Ukrainian SSR. A younger generation of his students include
V.~M.~Adamyan, D.~Z.~Arov, G.~Langer, F.~E.~Melik--Adamyan,
I.~E.~Ovcharenko, Sh.~N.~Saakyan, I.~M.~Spitkovskii, V.~A.~Avryan,
and others.

M.~G.~Krein is an author of more that 300 papers and monographs all
of which with no exception were published abroad several times.
These works are of an excellent analytic level and quality, broad in
topic, and have opened a number of new directions in mathematics,
while significantly enriched traditional directions. They initiated
and continue to inspire many mathematicians, engineers, physicists
throughout the world.  The following is an incomplete list of
branches of mathematics where M.~G.~Krein's research became
fundamental and, to an extend, have determined the direction for a
later development: oscillation kernels and matrices, the moment
problem, orthogonal polynomials and approximation theory, cones and
convex sets in Banach spaces, operators on spaces with two norms,
extension theory for Hermitian operators, the theory of extension of
positive definite functions and spiral arcs, the theory of entire
operators, integral operators, direct and inverse spectral problems
for inhomogeneous strings and Sturm--Liouville equations, the trace
formula and the scattering theory, the method of directing
functionals, stability thory for differential equations,
Wiener--Hopf and Toeplitz integrals and singular integral operators,
the theory of operators on spaces with an indefinite metric,
indefinite extension problems, non-selfadjoint operators,
characteristic operator-valued functions and triangular models,
perturbation theory the Fredholm theory, interpolation theory and
factorization theory, forecast theory for stationary stochastic
processes, problems in elastic theory, the theory of vessel waves
and wave resistance. A characteristic feature of his works is a deep
inner unity, an interlacing, of abstract and geometric ideas with
concrete analytic results and their applications. Since the range of
mathematical interests of Marko Grygorovych is very broad, let us
consider only the main, to our opinion, directions of his research.

An important role in the development of functional analysis and its
application is played by M.~G.~Krein's papers on geometry of Banach
spaces and linear topological spaces, and operators that act on
them. Here, first of all, let us stress on the introduction and a
study of Banach spaces with a fixed cone of vectors and dual to
them, spaces with two norms, convex sets and weak topologies in
Banach spaces. Very famous became the Krein--Mil'man theorem on
extreme points of convex sets, and the Krein--Kakutani theorem on
an isomorphism of an abstract Banach space with identity and endowed
with a vector structure to the space of continuous functions on a
compact Hausdorff space.

A unification of algebraic and geometrical methods is clearly seen
in a study of Marko Grygorovych in the theory of topological groups
and homogeneous spaces. Harmonic analysis on a locally compact
commutative group and finding a duality-like principle for compact
noncommutative groups (for commutative groups, the dual object
becomes the group of characters), in particular, the fact that the
structure of a homogeneous compact spaces is completely determined
by the algebra of harmonic functions had a significant impact to the
subsequent development of the abstract harmonic analysis.

M.~G.~Krein has completely described positive selfadjoint extensions
of positive selfadjoint operators and developed their
classification. An essential role here is plaid by two extreme
extensions, the strict (Friedrichs extension) and a soft extension,
later called the Krein extension. These results were applied to a
study of boundary-value problems for ordinary differential
equations. Using and expanding the theory of analytic functions, he
studied Hermitian operators with equal deficiency indices and found
a new interesting class of operators, he called entire, that
admitted for analogues of main constructions in the classical moment
problem in the undetermined case. This theory permitted to connect
the following problems, distinct at a first glance: a moment
problem, the problem of continuation of positive definite functions
and spiral arcs, a description of spectral functions of a string,
and others, and, in some sense, this permitted to solve these
problems completely. This theory has lead to new orthogonal problems
in the theory of analytic functions, which were solved, and once
more demonstrated the prediction of M.~G.~Krein that said that
``significant successes in functional analysis can be achieved by
broadening the modern machinery of analytic function theory and, at
the same time, functional analysis poses problems and stimulates a
development of the theory''.

M.~G.~Krein has developed a general method of directing functionals
with a use of which he obtained an eigen function decomposition of
ordinary selfadjoint differential operators. This has extended the
many-year research of J.~Sturm, J.~Liouville, V.~A.~Steklov,
H.~Weyl on second order equations to differential equation of an
arbitrary order. This theory made a basis for developing a theory of
integral representation of positive definite kernels in terms of
elementary ones. As consequences, this theory gave well known
theorems of S.~Bokhner, S.~N.~Bernstein, etc, on integral
representations of functions that are positive definite,
exponentially convex, and other classes of functions. Here again,
Marko Grygorovych has shown his extraordinary ability to see behind
almost any concrete problem ``an impressive thing, some selfadjoint
unbounded operator'' such that its spectral decomposition solves the
problem.

For many years, Marko Grygorovych was putting many efforts working
on problems connected with stability of solutions of differential
equations, although he never regarded himself a specialist in this
field. He was saying that this was a hobby for him. The theory of
stability regions, developed by A.~M.~Lyapunov for second order
equations, was finally generalized by M.~G.~Krein, after a 50 year
intermission due to serious problems, to canonical systems with
periodic coefficients using methods of functional analysis. A
foundation of stability theory he created for differential equations
in a Banach space has permitted to achieve this goal in a simpler
way and, sometimes, in a more complete form even in the case of a
system with one degree of freedom.

M.~G.~Krein has made a significant contribution in the theory of
inverse problems for a Sturm--Liouville equation, a more general
equation of a string, and canonical systems of differential
equations. In particular, he has solved the problem of
reconstructing a Sturm--Liouville equation from two spectrums, and a
canonical system from its spectral function or from a scattering
matrix. That used the analytic machinery developed when studying
entire operators and the theory of systems of Wiener--Hopf
equations.  The state of the theory at the time was not satisfactory
for M.~G.~Krein, and he succeeded to go far ahead in constructing a
general theory of such systems. This theory has achieved perfection
and completeness in a number of his publications, and he was awarded
the M.~M.~Krylov Prize in 1979. The theory was constructed on the
basis of factorization of matrix-valued functions.  Factorization
problems for functions, matrix- and operator-valued functions were
always interesting to M.~G.~Krein by themselves. Let us also mention
that, in the coarse of these studies, a theory of accelerants has
appeared, a theory that can be considered in the case of canonical
systems with two unknown functions as a continual analogue of
orthogonal polynomials on the circle. Developing further methods
that he proposed for solving the inverse spectral problem for a
string, M.~G.~Krein and his students have solved the problem of
reconstructing a string, possibly singular, with friction at one
end, by using a sequence of eigen frequencies and, as well as
function theory problems related to such a string, and considered
the existence problem for a special representation of a polynomial
that is positive on a system of closed intervals. This problem and
also the extremal problems for polynomials, which he solved,
generalize the corresponding problem of A.~A.~Markov who worked with
only one interval.

M.~G.~Krein's ideas and methods have also deeply penetrated the
theory of non-selfadjoint operators. They helped this theory, which
was considered in the talk of M.~G.~Krein at the congress in Moscow
in~1966 as one of links of a ``certain connected set of events
taking place in the area of Hilbert spaces'', to look nowadays as a
real mountain chain that has its own architecture, its analytic
tools, and one can say that a special calculus that has unexpected
applications in different areas of analysis. So, here again,
M.~G.~Krein has achieved his ``Cape of Good Hope'', and we can say
that the ``Cape of Previsions''.

M.~G.~Krein was one of creators of the theory of operators on spaces
with indefinite metric. His idea of a defining polynomial and the
method of defining functionals made a foundation for the theory of
integral representations and continuation of Hermitian-indefinite
functions with a finite number of negative squares, the theory of
spectral decompositions of selfadjoint and unitary operators on
$\Pi_{\kappa}$-type Pontryagin spaces, which now has achieved the
level of development similar to the one of the corresponding theory
on a Hilbert space. The geometry of spaces named after M.~G.~Krein
and operators on these spaces attract increasingly more attention of
both theorists and practicians. On the basis of the obtained
results, one has studied generalized classes of Schur functions,
Caratheodori functions, Nevanlinna functions, which are generalized
in the sense that the quadratic forms connected with them have a
finite number of negative squares. Using these classes, one can
define corresponding generalizations of classic discrete and
continual problems, which are trigonometric and power moment
problems, the Schur problem, the Nevanlinna--Pick problem, and
others. For this case, one develops the problems considered for the
definite case, that are a theory of accelerants, continual analogues
of orthogonal polynomials, spectral theory of canonical systems. A
next step is a continual version of Nehari's problem for rectangular
contracting matrix-valued functions on the real axis, and an
application for solving matrix-continual analogs of the Schur
problem and the Caratheodori--Toeplitz problem.

In the problems described above, and in other problems of harmonic
analysis, a description of solutions of the problem is given in
terms of a fractional-linear transformation over a class of
contracting analytic matrix-valued functions such that the
coefficient matrix has certain properties. This formula became a
starting point for finding extremal value solutions of a so-called
entropy functional that plays a special role in a number of
applications.

Close ties between theoretical and applied aspects in the work of
M.~G.~Krein reflected in multiple applications of his results in
numerous branches of science and technology. As we have already
mentioned, his studies of the moment problem were connected with
optimal control problems with distributed parameters, the theory of
extensions of positive definite functions was related to linear
predictions for stationary processes, the proposed method for
determining critical frequencies in the parametric resonance
phenomenon is used in the synchrophasotron theory. Let us also
recall, in this connection, his results in the theory of vessel
waves and wave resistance. His method of calculating the number of
negative eigen values of an extension of a positive Hermitian
operator is used for studying stability of structures. Contact
problems in elasticity theory, the theory of molecular interactions
also use results of Marko Grygorovych. His studies of topological
groups were recently applied in the graph theory, and the Krein
algebras are used in modern combinatorial analysis. One should also
recall a number of works that developed Krein's ideas, written
together with his students, on infinite dimensional Hankel matrices
and the generalized Schur problem (Nehari's problem), which
gave an impulse to a new direction in the control theory, the
$H_{\infty}$-optimal control. Nowadays it is a subject of many
papers, monographs, conferences.

M.~G.~Krein was not only a prominent scientist but an excellent
pedagogue. He have taught many world famous scientists among whom
there are 20 Doctors of Sciences and 50 Doctor Candidates. He
generously shared his knowledge and plans with them as well as with
his other colleagues. For more than half a century, M.~G.~Krein have
been conducting a City Mathematical Seminar that was conducted in
the House of Scientists in Odessa for a long time, and then in the
Institute of Civil Engineering, and still later in the South Science
Center.  Older and younger scientists, Krein's students and friends,
were participating in the work of the Seminar. They included
V.~M.~Adamyan, D.~Z.~Arov, M.~P.~Brodskii, Yu.~P.~Ginzburg,
I.~Ts.~Gohberg, G.~M.~Gubarev, I.~S.~Iohvidov,
I.~S.~Kats, K.~R.~Kovalenko, G.~Langer, F.~E.~Melik-Adamyan,
S.~M.~Mkhitaryan, A.~A.~Nudel'man, I.~E.~Ovcharenko, G.~Ya.~Popov,
Sh.~N.~Saakyan, L.~A.~Sakhnovich, I.~M.~Spitkovskii,
Yu.~P.~Shmul'yan, V.~A.~Yavryan. Making a talk in this seminar was
considered an honor for a mathematician in the former Soviet Union.
Also M.~G.~Krein was conducting smaller seminars in institutes where
he worked. In the Odessa Marine Institute, he have organized a
seminar on hydrodynamics. Among the participants there were
Yu.~L.~Vorob'iev, A.~A.~Kostyukov, V.~G.~Sizov. In the Kuibyshev
Industrial Institute, there was a seminar, which he headed, with
G.~Ya.~Lyubarskii, O.~V.~Svirskii, A.~V.~Shtraus participating in
its work. Also, as was already mentioned, while working in the
Institute of Mathematics of the Academy of Scines of the Ukrainian
SSR, he headed a seminar on functional analysis, where also worked
Yu.~M.~Berezansky, Yu.~L.~Daletsky, G.~I.~Kats, B.~I.~Korenblyum,
M.~O.~Krasnosel'skii, S.~G.~Krein. Almost each year, Marko
Grygorovych gave a coarse of lectures to students and young
researchers, based on his new results. Many of them have remain
still unpublished. Only in 1997, notes of his lectures on the theory
of entire operators, given at Odessa Pedagogical Institute, were
made available by V.~M.~Adamyan and D.~Z.~Arov, prepared and
enlarged by V.~I.~Gorbachuk and M.~L.~Gorbachuk, and published by
Bikh\"a{}user with a help of I.~Ts.~Gohberg. A similar thing has
happened with his course of lectures given at Moscow State
University, where he discussed his results on the theory of
forecasting for many dimensional stochastic processes, where
Yu.~A.~Rozanov was one of participants. Later his notices became a
part of his review published in ``Uspekhi Matematicheskikh Nauk''.
A course of lectures given by Marko Grygorovych Krein at USSR
mathematical schools, namely, ``On some new studies of perturbation
theory'' (Kanev, 1963), ``An introduction to geometry of indefinite
$J$-space and a theory of operators on it'' (Katsiveli, 1964) left
an unforgettable impression with its depth and the number of posed
problems. At the International Mathematical Congress (Moscow, 1966),
his one-hour talk ``Analytic problems and results of the theory of
operators on a Hilbert space'' was followed with loud applause from
an overcrowded large auditorium, to what L.~V.~Kantorovich, the head
of the session, have remarked that even most famous actors not
always get such an ovation.

Marko Grygorovych was a benevolent, fair person, although exigent to
others and himself. The level of his scientific ethics is shown by
the following example. When studying entire operators with the
deficiency index $(1,1)$, it is important to consider the resolvent
matrix, the use of which gives a description of all spectral
functions of such operators. M.~G.~Krein has shown that this matrix
is a monodromy matrix of a certain canonical system, and made a
hypothesis that there is a unique Hamiltonian of this system, with a
certain normalization, which he proved only for positive operators.
In the general case, this was proved by Louis de Branges with a use
of functional but not operator methods. During one of his talks at a
meeting of the Moscow Mathematical Society, Marko Grygorovych gave
the following account of the work of Louis de Branges: ``I consider
it brilliant. In a short time he (Louis de Branges) have succeeded
in finishing the distance that took me so many years.  Louis de
Branges have repeated many of my propositions but the final result
belongs to him. I was heading to it but never reached it''.  From
the ethic point of view, these words can be compared with those of
Euler about solving the isoperimetrical problem by Lagrange. In a
letter to Lagrange, he wrote ``Your analytic solution of the
isoperimetrical problem contains everything that one can wish. I am
quite happy that the theory I work on almost by myself is brought by
you to the highest level of perfection''.

Scientific achievements of M.~G.~Krein were widely acknowledged by
the international mathematical community. He was one of four Soviet
mathematicians who was elected a foreign member of the American
Academy of Arts and Sciences, a member of the US National Academy of
Sciences, a member of many mathematical societies and editorial
boards of leading mathematical journals. In 1982, Marko Grygorovych
was awarded with the International Wolf Prize in Mathematics, which
is an analogue of the Nobel Prize in mathematics.  The foreword to
the prize contains the words ``his achievements are a culmination of
the famous direction started by Tchebycheff, Stieltjes,
S.~Bernstein, and Markov, and continued by F.~Riesz, Banach, and
Szego. Krein has achieved to apply powerful methods of
functional analysis to problems in the theory of functions,
probability theory, and mathematical physics. His research has lead
to a sensible increase of applications of mathematics to various
fields from theoretical mechanics to electrical engineering and
control problems. His mathematical style, personal leadership, and
purenes, have laid standards of outermost mastership''. One of the
best books of well-known American mathematicians P.~Lax and
R.~Fillips ``Scattering theory for automorphic functions" (Princeton
University Press and University of Tokyo Press, 1976) was dedicated
to Marko Grygorovych, ``one of the giants in Mathematics of 20-th
century, as an homage to its extraordinarily broad and deep
contribution to mathematics''.

Regardless all this, his scientific career in his own country, we
have already said, was finished already in 1939. Accused in Jewish
nationalism and cosmopolitanism, that he cited foreign authors very
often and conversely, of being cited by foreign mathematicians
(Marko Grygorovych is one of most cited authors in the world), he
never became a member of the Academy of Sciences of the Soviet Union
nor of the Academy of Sciences of the Ukrainian SSR. It may seem
that the required standards were too high for M.~G.~Krein. Many of
his students could not obtain, from the Highest Attestation
Committee, a confirmation of their PhD degree. Multiple proposals by
the Moscow Mathematical Society and other influential mathematical
institutions and individual mathematicians such as
P.~S.~Aleksandrov, A.~N.~Kolmogorov, I.~G.~Petrovsky for awarding
Marko Grygorovych the State Prize or any other prestigious prize
always ended, regardless all their arguments, by excluding the name
of M.~G.~Krein from the list of the candidates for the prize. No
state or governmental institution would support him so the
presidents of neither the Academy of Sciences of the Soviet Union
nor the Academy of Sciences of the Ukrainian SSR would do anything
for him, although they were well aware of the level of his research.
The President of the Academy of Sciences of the Soviet Union
M.~V.~Keldysh would ask the President of the Academy of Sciences of
the Ukrainian SSR B.~E.~Paton why the most famous mathematician of
Ukraine is still not a member of the Academy, although B.~E.~Paton
could address the same question to M.~V.~Keldysh.

He could not leave the Soviet Union. During all his life he never
crossed the border of the country. He even could not receive the
Wolf Prize. One time he obtained a permission to take part in a
conference in Hungary (in Balatone), however he has not used the
visa for there was the cholera disease in Odessa at this time and he
could not leave the city for sanitary reasons. When I.~Ts.~Gohberg,
who participated in the conference, informed B.~Sz.-Nagy, the head
of the Organizing Committee, about the reason for M.~G.~Krein not
coming to the conference, knowing the attitude of the officials to
the subject he said with a smile, --- ``So now it is called cholera
?''. That was the only time when the reason of M.~G.~Krein's absence
was true. Also, no foreign scientist who wished to meet him could
travel to Odessa. This happened, for example, to J.~Helton and
R.~Phillips. All this was explained that there is no branch of the
Academy of Science in Odessa.

During the Perestroika times, the situation changed to a better.
Marko Grygorovych, together with M.~M.~Bogolyubov, was awarded the
State Prize in the area of Science and Technology in 1987. During
the years of independence, the Institute of Mathematics of the
National Academy of Sciences of Ukraine publishes a three volume
collection of his works that appeared in not easily accessible
journals. In 2006 the Presidium of the National Academy of Sciences
of Ukraine has decided to found the M.~G.~Krein Prize to be given
for distinguished achievements in functional analysis. Only it is a
pity that Marko Grygorovych did not see how a country called the
USSR has disappeared from geographical maps and Ukraine became
independent. He died on October 17, 1989, and did not see the
Pamoranchova Revolution. It is certain that, in his thoughts, he
would be in the Square, together with his daughter Irma, with his
``scientific children and grand children''.

However, regardless all the problems he has experienced in his time,
Marko Grygorovych was a happy person, since happiness is granted
only to those who know a lot, and the more the person knows the
stronger and distinctly he sees the poetry in the world where one
with a poor knowledge may never find it. Looking into a puddle in
the dusk some people see the water, other see the stars.  Marko
Grygorovych saw the stars. We are happy to be his contemporaries.\\
[5mm] V.~M.~Adamyan, D.~Z.~Arov, Yu.~M.~Berezansky, V.~I.~Gorbachuk,
M.~L.~Gorbachuk, V.~A.~Mikhailets, A.~M.~Samoilenko








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