Mikhail Krein

Mikhail Krein was a Soviet mathematician noted for foundational work in functional analysis, operator theory, and spectral theory. He developed methods linking positive definite functions, moment problems, and extension theory of symmetric operators, influencing generations of researchers at institutions such as Moscow State University, the Steklov Institute of Mathematics, and international centers in Paris, Cambridge, and New York City. His work intersected with contemporaries including Israel Gelfand, Naum Akhiezer, Mark Krein, Frigyes Riesz, and John von Neumann.

Early life and education

Born in Poltava in the Russian Empire, he received early schooling that led him to Moscow State University, where he studied mathematics during the tumultuous years after the Russian Revolution of 1917 and amid societal transformations under Soviet Union policies. At Moscow State University he encountered lectures and seminars connected to figures from the St. Petersburg School and the emerging Leningrad Mathematical School, absorbing influences from teachers associated with the Steklov Institute of Mathematics and scholars linked to Vladimir Andreevich Steklov and Dmitri Egorov. His formative education brought him into contact with problems studied by researchers at the Academy of Sciences of the USSR and with literature produced by authors in the Mathematical Institute of the USSR Academy of Sciences.

Career and academic positions

He held positions at Moscow State University and at the Steklov Institute of Mathematics, participating in seminars that connected to the Moscow Mathematical Society and to international exchanges involving the International Congress of Mathematicians. During his career he collaborated with mathematicians from the Institute for Advanced Study and institutions in France, Germany, and the United States, building links between Soviet analysis schools and research centers such as Harvard University, University of Cambridge, and Princeton University. He supervised research in departments associated with the Moscow State University Department of Mechanics and Mathematics and contributed to organizational efforts within the Soviet Academy of Sciences and editorial work for journals influenced by the Russian Mathematical Surveys tradition.

Mathematical contributions and research

His research advanced aspects of operator theory by developing extension theories for symmetric and Hermitian operators, refining approaches that related to the moment problem studied earlier by Thomas Stieltjes and Gustave Choquet. He produced important results on positive definite kernels, connecting to the legacy of S. Bochner and Norbert Wiener, and on the spectral analysis of non-self-adjoint operators in contexts related to the Krein–Milman theorem lineage and to work by M. G. Krein and I. M. Gelfand. Krein's techniques addressed inverse spectral problems that linked to the Sturm–Liouville theory and to classical problems in harmonic analysis developed by Andrey Kolmogorov and Nikolai Luzin.

He formalized operator models that clarified the structure of linear relations and dissipative operators, drawing upon concepts previously explored by John von Neumann and later extended in the frameworks used by T. Ya. Azizov and I. S. Iohvidov. His work on moment problems illuminated connections between analytic function theory from the Russian function theory school and operator extensions associated with the Hilbert space methods propagated by Frigyes Riesz and Stefan Banach. Krein's papers influenced spectral theory for differential operators, resonating with contributions by Harald Weyl and linking to mathematical physics problems addressed at the Lebedev Physical Institute and in studies related to quantum mechanics foundations by Paul Dirac and Eugene Wigner.

Students and legacy

He mentored a cohort of students who later became prominent in functional analysis and mathematical physics, many occupying chairs at Moscow State University, the Steklov Institute of Mathematics, and international departments in France and the United States. His academic descendants propagated techniques for inverse problems, interpolation theory, and the study of generalized resolvents, contributing to schools associated with Naum Akhiezer, Israel Gelfand, and Mark Krein. Through seminars of the Moscow Mathematical Society and collaborations with researchers at the Institute for Low Temperature Physics, his legacy shaped later work on operator models applied in engineering contexts related to control theory and to applied studies in signal processing undertaken at institutions like Moscow Power Engineering Institute and universities in Germany.

Awards and honors

During his lifetime he received recognition from bodies within the Soviet Union and from international scientific societies connected to the International Mathematical Union and to academies such as the USSR Academy of Sciences and foreign academies in France and Poland. His publications became standard references cited alongside monographs by Israel Gelfand, Naum Akhiezer, and Mark Krein, and his name appears in eponymous theorems and methods discussed in seminars of the Moscow Mathematical Society and at conferences such as the International Congress of Mathematicians.

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