On 7 March 2011 Mikhail Iosifovich Kadets
passed away.
He was born in Kiev on 30 November 1923.
His father, Iosiph Mikhailovich, was arrested in 1937 as "Enemy of the People" and has died in Stalin's camps. After graduating from school and serving in
the army, he enrolled in 1946 in the Faculty
of Physics and Mathematics at Kharkov
State University. While still a student, Kadets
became interested in Banach space theory. In
1948 Banach’s famous monograph “Th´eorie des
op´erations lin´eaires” (1932) was published in
Ukrainian. Kadets studied this book deeply
and was fascinated by many of the problems
formulated there. This monograph became his
reference book, and the problems stated by
Banach remained the subject of Kadets’ interests
for the rest of his life.
After graduating from the university Kadets moved from Kharkov to Makeevka
in Donetsk Province, where he worked for seven years in the Research Institute
of the Coal Mining Ministry and taught mathematics and physics at a firefighting
technical school. He was never a postgraduate student and had no scientific advisor
in the standard sense of the word.
In Makeevka he obtained his first interesting research results, and in 1955 in
Kharkov he defended his Ph.D. dissertation written on the basis of those results.
In 1957 he returned to Kharkov and from that time taught there in various technology
institutes of higher education. Since 1964 and until his death
he worked in Kharkiv State Academy
of Municipal Economy
http://www.kname.edu.ua/. In 1963 he defended his D.Sc. dissertation. His publications from that period were concerned with two areas: a topological
classification of Banach spaces, and vector series. Numerous problems which arose
there became the topics of his subsequent investigations.
After 12 years of intense work in the first area, Kadets solved in the affirmative
the longstanding Fr´echet–Banach problem of homeomorphism of all separable
infinitedimensional Banach spaces. This remarkable result immediately became
classical.
One of his tools in the solution of this problem was the construction of equivalent
norms satisfying some special convexity conditions. It then turned out that the
technique of equivalent norms was effective in a much broader range of problems in
the geometry of Banach spaces and nonlinear analysis. Kadets is justly regarded as
one of the creators of the theory of equivalent renormings of Banach spaces, which
has now grown into a separate area.
His main results in the theory of series relate to infinitedimensional analogues of
Steinitz’ famous theorem on rearrangements. The latter is a finitedimensional analogue
of Riemann’s classical theorem on conditionally convergent numerical series. For series in Lp Kadets found sufficient conditions on the rate of decrease of the
terms of the series ensuring that the set of sums of its convergent rearrangements
is a translation of some subspace. Later on, E. M. Nikishin and P. A. Kornilov
showed that these conditions are also necessary. But without any additional conditions
the set of sums can have a quite different structure. In a joint paper Kadets
and K.Wo´zniakowski proved (1989) that in any infinitedimensional Banach space
there is a series such that the set of sums of its convergent rearrangements consists
of two points.
Of great importance for the development of the theory of series in Banach spaces
was the book “Series in Banach spaces: conditional and unconditional convergence”
(Birkh¨auser, 1997) written by M. I. Kadets together with V. M. Kadets.
More than 60 years have now passed since the publication of the Ukrainian
translation of Banach’s book. The appearance of Banach space theory has changed
significantly since then, in many respects thanks to Kadets’ results. Almost all of
them were subsequently developed further in one way or another, and some of them
underlie whole areas of modern Banach space theory. Here are several examples.
His result that a separable Banach space has an equivalent Fr´echetdifferentiable
norm if and only if the dual space is separable, became one of the basic results of
the theory of smooth renormings (see the monograph by R. Deville, G. Godefroy,
and V. Zizler, Smoothness and renormings in Banach spaces, Wiley, 1993).
The Kadets–Snobar estimate for the projection constant of an ndimensional
normed space and the Gurarii–Kadets–Macaev (Matsaev) estimate of the Banach–
Masur distance between the ndimensional lp1 and lp2 spaces are significant contributions
to the theory of finitedimensional spaces (see N. TomczakJaegermann’s
monograph: Banach–Mazur distances and finitedimensional operator ideals, Wiley,
1989).
The Kadets–K¨ursten theorem that a vectorvalued function which is almost periodic
on each linear functional must have a countable spectrum and the theorem of Kadets on the maximal class of Banach spaces in which the Bohl–Bohr theorem on
the integral of an almost periodic function holds are central results in the theory of
vectorvalued almost periodic functions.
The fundamental results due to Kadets and A. Pe lszy´nski on the linear topological
structure of the Lpspaces have numerous applications not only in Banach
space theory but also in the theory of functions.
Kadets’ theorem that there exists a separable Banach space E such that each
separable Banach space with the bounded approximation property is isomorphic to
a complemented subspace of E is one of the central results in the theory of universal
Banach spaces.
His 1/4 theorem solved the famous Paley–Wiener problem on the basis property
for systems of exponentials in L2.
Mikhail Iosifovich Kadets devoted much time and effort to his pedagogical activities. Nineteen
of his students defended their Ph.D. dissertations and among them seven became
doctors of the sciences. He was generous in sharing his mathematical ideas with
his students. His Kharkov school was well known internationally. In particular,
A. Pietsch mentions it as a phenomenon in Banach space theory in his book “History
of Banach spaces and linear operators” (Birkh¨auser, 2007).
In 1991 Kadets was given the title of Honoured Scientist of Ukraine, and in 2005
he was awarded a State Prize of Ukraine.
Mikhail Iosifovich Kadets was a brilliant and an exceptionally deep mathematician,
a kind and sympathetic person, witty and pleasant to talk with. That is how
he will remain in our thoughts.
