Mark Krasnoselskii

Mark Krasnoselskii
Name	Mark Krasnoselskii
Native name	Марк Аронович Красносельский
Birth date	1920
Death date	1997
Nationality	Soviet Union
Fields	Mathematics
Institutions	Ural State University; Moscow State University; Steklov Institute
Doctoral advisor	Nikolai Luzin

Mark Krasnoselskii was a Soviet mathematician known for foundational work in nonlinear functional analysis, operator theory, and the theory of nonlinear integral equations. He developed techniques that connected topological methods from the work of Lefschetz fixed-point theorem and Brouwer fixed-point theorem with analytic approaches related to Hilbert space methods and Banach space theory. Krasnoselskii's results influenced research in applied fields linked to the work of Schauder, Krasnoselskii, and contemporaries such as Mikhail Lavrentyev and Israel Gelfand.

Biography

Born in the Soviet Union in 1920, Krasnoselskii studied under Nikolai Luzin at Moscow State University and was active in the mathematical community centered on the Steklov Institute of Mathematics and Ural State University. His career unfolded during periods marked by the influence of figures like Andrey Kolmogorov, Pavel Alexandrov, and Lev Pontryagin, and he collaborated with mathematicians associated with Leningrad State University and researchers in Novosibirsk. Krasnoselskii supervised students who later worked at institutions including Moscow State University, the Steklov Institute, and universities across the Soviet Union and Eastern Europe. His academic life intersected with events such as developments in Soviet mathematics during the mid-20th century and contacts with scholars from France, Germany, and Japan.

Mathematical Contributions

Krasnoselskii made major advances in nonlinear analysis by synthesizing methods from the theory of compact operators on Banach space, topological degree theory associated with Brouwer fixed-point theorem and Leray–Schauder degree, and order-preserving operator techniques related to the work of G. Birkhoff and Emil Artin. He introduced and exploited cone-theoretic methods connected to Krein–Rutman theorem and extended spectral ideas resembling those in Fredholm theory and Riesz–Schauder theory. His approach brought tools from measure theory and functional analysis into contact with problems originating in boundary value problems and integral equations studied by researchers such as Israel Gelfand, Einar Hille, and Marshall Stone.

Major Theorems and Concepts

Krasnoselskii is associated with several named results and frameworks that permeate modern analysis, including techniques related to the Krasnoselskii fixed-point theorem, which combines compression-expansion mappings in cones and is often applied alongside the Schauder fixed-point theorem and Banach fixed-point theorem. He formulated versions of the Krein–Rutman theorem for nonlinear settings and developed comparison principles related to monotone operator theory leveraged by followers such as Tikhonov and Zabreiko. His work provided foundations for existence results in nonlinear integral and operator equations that influenced later results by Browder, Tao, and Zeidler.

Publications

Krasnoselskii authored several influential monographs and papers, including treatises on nonlinear operator equations, integral equations, and applications of topological methods. His books interacted with literature by S. Banach, J. Leray, J.-L. Lions, J. Schauder, and monographs circulated through institutions like the Steklov Institute. These publications were used as standard references in graduate courses at Moscow State University, Ural State University, and translated for use in mathematical centers in United States, France, and Japan where scholars such as Haim Brezis and Eberhard Zeidler referenced his methods.

Awards and Honors

During his career Krasnoselskii received recognition from bodies tied to the Soviet scientific establishment and was honored in conferences that included participants from Moscow State University, the Steklov Institute, and international symposia connected to organizations like the International Mathematical Union and national academies including the Russian Academy of Sciences. Colleagues commemorated his work through dedicated sessions at meetings associated with institutions such as Lomonosov Moscow State University and publications appearing in proceedings alongside contributions by Kolmogorov, Pontryagin, and Gelfand.

Legacy and Influence

Krasnoselskii's methods remain central in contemporary studies of nonlinear problems treated in texts influenced by authors like Haim Brezis, Eberhard Zeidler, Murat, and Rabinowitz. His fixed-point techniques and cone-theoretic perspectives underpin modern research in areas pursued at centers such as Princeton University, University of Cambridge, Institut Henri Poincaré, and RIMS in Kyoto, and inform applied analysis in works from researchers at ETH Zurich, CNRS, and California Institute of Technology. Graduate curricula in functional analysis and seminars at institutions including Moscow State University continue to teach Krasnoselskii-inspired methods, and his theorems are cited across literature on nonlinear differential equations, integral equations, and dynamical systems studied by scholars like Henry Poincaré and Jean Leray.

Category:Soviet mathematicians