(27.04.1920 --- 13.02.1997)

Mark Mark Alexandrovich Krasnosel'skii, renowned for his farreaching and innovative work on nonlinear functional analysis and its applications, died while working at his desk in Moscow on the 13th of February 1997, just 6 weeks short of his 77th birthday. He had spent parts of 1995 and 1996 as a Humboldt-Preis-Tr"ager at the Universit"at Regensburg and the Weierstrass--Institut in Berlin. His ideas and results have had a profound impact on mathematics and its applications throughout the international mathematical community and in Germany in particular.
Mark Krasnosel'skii was born on the 27th of April 1920 in the town of Starokonstantinov in the Ukraine where his father worked as a construction engineer and his mother taught in an elementary school. In 1932 the Krasnosel'skii family moved to Berdyansk and in 1938 entered the physico-mathematical faculty of Kiev University, which was evacuated at the beginning of World War II to Kazakhstan where it became known as the Joint Ukrainian University. After graduating in 1942 Mark Krasnosel'skii spent the next four years as an artillery officier in the Soviet Army and in August 1946 he returned to Kiev to commence his graduate studies while holding a junior position in the Mathematical Institute of the Ukranian Academy of Sciences. In 1948 Krasnosel'skii defended his Kandidat of Science (equivalent to Ph.D.) dissertation on extension theory for Hermite operators and in 1950 he completed his Doctor of Science (similar to Habilitation) dissertation on topological methods in nonlinear analysis. In 1953 he was appointed to a Chair in Functional Analysis at the Voronezh University and during the 15 years that he stayed there his seminar on nonlinear analysis became well-known far beyond Voronezh. In 1968 Krasnosel'skii moved from Voronezh to Moscow where he worked at the USSR Academy of Sciences, first at what is now called the Institute of Control Sciences and then from 1990 until his death at the Institute for Information Transmissions Problems.
In over a half century of scientific activity Mark Krasnosel'skii wrote more than 380 scientific articles and 14 monographs. He was an indefatable hard worker, an outstanding teacher, an energetic scientific organisor and an inspiration to all those who came into contact with him. Throughout his scientific career he worked and published extensively with many other mathematicians and scientists in Russia and further a field. In addition, 33 of his students completed the Doctor of Science degree. His scientific interests were extensive and covered many aspects of modern mathematics. In particular, he was responsible for opening up important new mathematical directions, the development of which created the foundations of modern nonlinear analysis.
Mark Krasnosel'skii was the first to investigate the functional analytical properties of fractional powers of operators, at first for self-adjoint operators and then for more general situations. His theorem on the interpolation of complete continuity of such fractional power operators has been a basic tool in the theory of partial differential equations. Of comparable importance in applications is his extensive collection of works on the theory of positive operators, in particular results in which spectral gaps were estimated. His work on integral operators and superposition operators has also found many theoretical and practical applications. A major reason for this was his ceaseless desire always to find readily verifiable conditions and estimates for whatever functional properties were under consideration. This is perhaps best seen in his work on topological methods in nonlinear analysis which he developed into a universal method for finding answers to such qualitative problems such as evaluating the number of solutions, describing the structure of a solution set and conditions for the connectedness of this set, convergence of Galerkin type approximations, the bifurcation of solutions in nonlinear systems, and so on. Krasnosel'skii also presented many new general principles on solvability of a large variety of nonlinear equations, including one-sided estimates, cone stretching and contractions, fixed-point theorems for monotone operators and a combination of the Schauder fixed point and contraction mapping theorems that was the genesis of condensing operators. He suggested a new general method for investigating degenerate extremals in variational problems and developed qualitative methods for studying critical and bifurcation parameter values based on restricted information of nonlinear equations such as the properties of equations linearized at zero or at infinity, which have been very useful in determining the existence of bounded or periodic solutions. After he moved to Moscow he turned his attention increasingly to discontinuous processes and operators, in connection firstly with nonlinear control systems and then with a mathematically rigorous formulation of hysteresis which encompasses most classical models of hysteresis and is now standard. He also became actively involved with the analysis of desynchronized systems and the justification of the harmonic balance method commonly used by engineers. This summary of Mark Krasnosel'skii's mathematical contributions is but a brief and selective sketch. A more thorough appreciation is warranted and will undoubtedly appear in the future.
Mark Krasnosel'skii was of noble personality, straightforward and uncompromising in matters of principle but always open minded and ready to help everyone. He was a Big Man in both body and mind. He was a superb mathematician.

P.E. Kloeden
Weierstrass-Institut, Berlin
with assistance from
N.A.Bobylev, V.S.Kozyakin, A.M.Krasnosel'skii,
A.V.Pokrovskii and B.I.Sadovsky.