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| Kreĭn | |
|---|---|
| Name | Kreĭn |
| Birth date | 1907 |
| Death date | 1989 |
| Nationality | Soviet |
| Fields | Functional analysis; Operator theory; Differential equations |
| Alma mater | Saint Petersburg State University |
| Known for | Kreĭn spaces; extension theory; spectral shift function |
Kreĭn Kreĭn was a Soviet mathematician whose work shaped functional analysis, operator theory, spectral theory, mathematical physics, and the theory of differential operators. His contributions influenced researchers at institutions such as Steklov Institute of Mathematics, Moscow State University, Leningrad University, and collaborations with figures connected to Hilbert space methods and Pontryagin space theory. Through a combination of abstract operator-theoretic techniques and applications to concrete problems in Sturm–Liouville theory, Dirac operator analysis, and scattering theory, he left a lasting imprint on twentieth-century analysis.
Kreĭn was born in 1907 and pursued studies at Saint Petersburg State University and later worked at the Steklov Institute of Mathematics and Leningrad University, interacting with contemporaries associated with Andrey Kolmogorov, Israel Gelfand, Naum Akhiezer, Mark Krein (note: avoid using variants), and colleagues from the Soviet Academy of Sciences. He contributed to seminars linked to Pavlov Institute of Mathematics and maintained correspondence with analysts at Moscow State University and researchers connected to Ilya Piatetski-Shapiro, Georgii Pontryagin, Mikhail Livsic, and Boris Levitan. Throughout his career he engaged with problems that brought him into contact with authors publishing in venues tied to Izvestia Rossiiskoi Akademii Nauk, Matematicheskii Sbornik, and conferences related to International Congress of Mathematicians participants.
Kreĭn developed foundational results connecting abstract operator theory with concrete investigations of ordinary differential equations and integral equations. He introduced techniques that bridged ideas from Hilbert space frameworks to the indefinite inner-product settings of Pontryagin space research and influenced spectral analysis methods used in studies of the Sturm–Liouville problem, Schrödinger operator, and Dirac operator. His work addresses extension problems for symmetric operators, providing tools later used by scholars at institutions like Princeton University, University of California, Berkeley, Harvard University, and Oxford University who studied self-adjoint extensions, boundary conditions, and the spectral shift. He formulated principles that interact with results of John von Neumann, Marshall Stone, Frigyes Riesz, Norbert Wiener, and Israel Gelfand, and his ideas underlie developments in the theory of functional models promoted by Mikhail Livsic and the scattering frameworks studied by Mark Krein-inspired communities.
Kreĭn introduced and formalized the concept now known as Kreĭn spaces, a class of vector spaces with an indefinite inner product that generalizes Hilbert space geometry and connects to Pontryagin space constructions, enabling analysis of operators with nonpositive spectra. His formulation provided a setting for extensions of the Riesz representation theorem and adaptations of spectral theorems originally due to John von Neumann and Marshall Stone to indefinite metrics. These spaces became central to work by researchers at Tel Aviv University, University of Toronto, University of Copenhagen, and ETH Zurich who explored applications to indefinite Sturm–Liouville problems, operator colligations, and the mathematical underpinnings of relativistic quantum mechanics through links to Dirac operator analysis. Later developments by authors associated with University of Birmingham, Universität Karlsruhe, and University of Cambridge extended Kreĭn space techniques to the study of linear relations, boundary triples, and reproducing kernel spaces, connecting to names such as Vladimir Derkach, Henk de Snoo, and Friedrichs-type inequalities.
Kreĭn's extension theory for symmetric operators provided canonical parametrizations of self-adjoint extensions, influencing works on the spectral shift function, trace formulas, and scattering theory. His methods interplay with results by Weyl, Titchmarsh, Birman, Krein-related spectral shift investigations, and developments in inverse spectral problems advanced by Borg, Marchenko, and Gelfand–Levitan techniques. Applications of his operator-theoretic constructions appear in studies at CERN-adjacent mathematical physics groups, in analyses by researchers at Courant Institute of Mathematical Sciences, Institute for Advanced Study, and in modern treatments of boundary value problems used by scholars linked to SIAM publications. Kreĭn's ideas inform contemporary approaches to nonselfadjoint operators, transmission problems, and the study of stability in systems examined at Caltech, Princeton Plasma Physics Laboratory, and by teams in the European Mathematical Society network.
Kreĭn's legacy is reflected in the sustained use of his namesake concepts across texts and research articles appearing in journals like Annals of Mathematics, Duke Mathematical Journal, Transactions of the American Mathematical Society, and Russian Mathematical Surveys. His impact is cited in monographs authored by academics at Cambridge University Press, Springer-Verlag, and in lecture series at International Congress of Mathematicians-related venues. Institutions including the Steklov Institute of Mathematics, Moscow State University, and departments at University of Oxford and Tel Aviv University preserve seminars and memorial lectures that perpetuate his methods, while researchers such as Vladimir Derkach, Henk de Snoo, Mikhail Livsic, and others continue to expand the body of work he initiated. Category:20th-century mathematicians