Mark Krein

Mark Krein was a Soviet mathematician noted for foundational work in functional analysis, operator theory, and spectral theory with lasting influence on Banach space theory, integral equations, and applications in mathematical physics. His research produced results that connected abstract operator methods with concrete problems in differential equations, moment problems, and the theory of positive operators. Collaborations and interactions with contemporaries shaped major advances in 20th-century analysis and influenced generations at institutions across the Soviet Union and internationally.

Early life and education

Born in Kyiv in 1907, he studied at the Kyiv Polytechnic Institute and later at the National University of Kyiv-Mohyla Academy predecessor institutions under the mentorship of prominent analysts. During his formative years he interacted with figures from the Leningrad School and the Kharkiv mathematical school, encountering the work of Naum Akhiezer, Lev Landau, Israel Gelfand, and Mark Naimark. His doctoral training occurred in a milieu connected to the Ukrainian Academy of Sciences and the broader Soviet research network that included the Steklov Institute of Mathematics and the Moscow State University mathematical circles.

Academic career and positions

He held positions at Taras Shevchenko National University of Kyiv and later at the Institute of Mathematics of the National Academy of Sciences of Ukraine. Throughout his career he lectured at institutions connected to the All-Union Academy of Sciences and collaborated with scholars from the Moscow Mathematical Society, the Kiev Mathematical Society, and the Institute for Theoretical Physics. He supervised students who became part of the Russian Academy of Sciences and influenced visitors from the Princeton University, University of Cambridge, and Harvard University who studied operator methods and spectral problems. His professional network extended to colleagues at the Weizmann Institute of Science, ETH Zurich, and institutions in Warsaw and Paris.

Contributions to functional analysis and operator theory

He made foundational contributions to the theory of Hermitian operators, self-adjoint operators, and the structure of indefinite metric spaces, linking concepts from Pontryagin space theory, Krein space ideas, and spectral analysis. His work on extensions of symmetric operators engaged with results of John von Neumann, Marshall Stone, M. A. Naimark, and W. N. Everitt, while his studies of resolvent formulas and operator pencils connected to the research of Israel Gohberg, Markus Krein collaborators, and Matveev. He advanced the study of moment problems building on contributions by Thomas Stieltjes, H. S. Wall, and M. G. Krein-related literature, and his insights influenced methods used by Barry Simon and Michael Reed in quantum mechanics spectral theory. His papers explored relationships among Herglotz functions, Schur complements, and Toeplitz operators, intersecting with work by Otto Toeplitz, Gustav Doetsch, and Frigyes Riesz. Through results on positive and compact operators he impacted the theories used by Einar Hille, Emil Artin, and Israel Glazman in differential operators and boundary-value problems.

Selected publications and the Krein–Rutman theorem

He authored and coauthored monographs and articles on operator extensions, spectral multiplicity, and positivity of operators; these texts were influential among analysts at the Steklov Institute, Institute of Applied Mathematics, and university departments worldwide. A notable result associated with his name is the Krein–Rutman theorem on eigenvectors of compact positive operators, developed in the context of work by Marcel Riesz, Stanislaw Rutkowski, and contemporaries such as Birkhoff and Perron; the theorem generalizes the Perron–Frobenius theorem for matrices to infinite-dimensional settings and has applications in population dynamics, integral equations, and partial differential equations. His selected papers treated the spectral theory of non-self-adjoint operators, kernel methods for integral operators, and inverse problems related to Sturm–Liouville theory, engaging ideas from Vladimir Marchenko, Levitan, and Gelfand-Levitan frameworks. Influential writings were circulated among researchers at Cambridge University Press venues and cited by scholars at Princeton University Press and by authors of surveys at the American Mathematical Society.

Honors and awards

He received recognition from the Ukrainian Academy of Sciences and was honored within the Soviet Academy systems for contributions to mathematics. His standing in the Mathematical Reviews community and the International Mathematical Union-linked conferences reflected the impact of his theorems and monographs. Colleagues commemorated him in memorial sessions at the Moscow Mathematical Congress and in proceedings of the International Congress of Mathematicians and regional symposia in Lviv and Odessa.

Category:1907 births Category:1989 deaths Category:Soviet mathematicians Category:Functional analysts Category:Operator theorists