Mikhail Kadets

Mikhail Iosiphovich Kadets (Russian: Михаил Иосифович Кадец, Ukrainian: Михайло Йосипович Кадець, sometimes transliterated as Kadec, 30 November 1923 – 7 March 2011) was a Soviet-born Jewish mathematician working in analysis and the theory of Banach spaces.[1][2][3]

Mikhail Kadets
Born(1923-11-30)30 November 1923
Kiev
Died7 March 2011(2011-03-07) (aged 87)
Kharkov
CitizenshipUkraine
Alma materKharkov University
Known forBanach–Fréchet problem
Kadets ​1⁄4-theorem
Kadets–Snobar estimate
Scientific career
FieldsBanach spaces
harmonic analysis
Doctoral advisorBoris Levin

Life and work

Kadets was born in Kiev. In 1943, he was drafted into the army. After demobilisation in 1946, he studied at Kharkov University, graduating in 1950. After several years in Makeevka he returned to Kharkov in 1957, where he spent the remainder of his life working at various institutes. He defended his PhD in 1955 (under the supervision of Boris Levin), and his doctoral dissertation in 1963. He was awarded the State Prize of Ukraine in 2005.

After reading the Ukrainian translation of Banach's monograph Théorie des opérations linéaires,[4] he became interested in the theory of Banach spaces.[5] In 1966, Kadets solved in the affirmative the Banach–Fréchet problem, asking whether every two separable infinite-dimensional Banach spaces are homeomorphic. He developed the method of equivalent norms, which has found numerous applications. For example, he showed that every separable Banach space admits an equivalent Fréchet differentiable norm if and only if the dual space is separable.[6]

Together with Aleksander Pełczyński, he obtained important results on the topological structure of Lp spaces.[7]

Kadets also made several contributions to the theory of finite-dimensional normed spaces. Together with M. G. Snobar (1971), he showed that every n-dimensional subspace of a Banach space is the image of a projection of norm at most √n.[8] Together with V. I. Gurarii and V. I. Matsaev, he found the exact order of magnitude of the Banach–Mazur distance between the n-dimensional spaces ℓn
p and ℓn
q.[9]

In harmonic analysis, Kadets proved (1964) what is now called the Kadets ​1⁄4 theorem, which states that, if |λn − n| ≤ C < ​1⁄4 for all integer n, then the sequence (exp(iλnx))n ∈ Z is a Riesz basis in L2[-π, π].[10]

Kadets was the founder of the Kharkov school of Banach spaces.[6] Together with his son Vladimir Kadets, he authored two books about series in Banach spaces.[11]

Notes
  1. "In memory of Mikhail Iosifovich Kadets (1923–2011)". Zh. Mat. Fiz. Anal. Geom. (in Russian). 7 (2): 194–195. 2011. MR 2829617.
  2. Lyubich, Yurii I.; Marchenko, Vladimir A.; Novikov, Sergei P.; Ostrovskii, M. I.; Pastur, Leonid A.; Plichko, Anatolii N.; Popov, M. M.; Semenov, Evgenii M.; Troyanskii, S. L.; Fonf, Vladimir P.; Khruslov, Evgenii Ya. (2011). "Mikhail Iosifovich Kadets (obituary)". Russ. Math. Surv. 66 (4): 809. doi:10.1070/RM2011v066n04ABEH004756.
  3. Gelʹfand, I. M.; Levin, B. Ya.; Marchenko, V. A.; Pogorelov, A. V.; Sobolev, S. L. (1984). "Mikhail Iosifovich Kadets (on the occasion of his sixtieth birthday)". Russian Math. Surveys. 39 (6): 231–232. doi:10.1070/rm1984v039n06abeh003197. MR 0771114.
  4. The French original Banach, S. (1932). Theory of Linear Operations. Monografje Matematyczne I. Warszawa: Mathematisches Seminar der Univ. Warschau. JFM 58.0420.01. was translated as Banach, S. (1948). Course in functional analysis (in Ukrainian). Kiev: Radians'ka shkola.
  5. Ostrovskii & Plichko (2009, First page of preprint): Ostrovskii, M. I.; Plichko, A. M. (2009). "On the Ukrainian translation of Théorie des opérations linéaires and Mazur's updates of the "remarks" section" (pdf). Mat. Stud. 32 (1): 96–111. MR 2597043.CS1 maint: ref=harv (link)
  6. Pietsch, Albrecht (2007). History of Banach spaces and linear operators. Boston, MA: Birkhäuser Boston, Inc. p. 609. ISBN 978-0-8176-4367-6. MR 2300779.
  7. Beauzamy, Bernard (1985). "Chapter VI". Introduction to Banach spaces and their geometry. North-Holland Mathematics Studies. 68 (2nd ed.). Amsterdam: North-Holland Publishing Co. ISBN 0-444-87878-5. MR 0889253.
  8. Fabian, Marián; Habala, Petr; Hájek, Petr; Montesinos, Vicente; Zizler, Václav (2011). Banach space theory. The basis for linear and nonlinear analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. New York: Springer. pp. 320–323. ISBN 978-1-4419-7514-0. MR 2766381.
  9. Tomczak-Jaegermann, Nicole (1989). Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics. 38. Harlow: Longman Scientific & Technical. p. 138. ISBN 0-582-01374-7. MR 0993774.
  10. Higgins, John Rowland (1977). Completeness and basis properties of sets of special functions. Cambridge Tracts in Mathematics. 72. Cambridge-New York-Melbourne: Cambridge University Press. ISBN 0-521-21376-2. MR 0499341.
  11. Kadets, Mikhail I.; Kadets, Vladimir M. (1997). Series in Banach spaces: Conditional and unconditional convergence. Operator Theory: Advances and Applications. 94 (Translated by Andrei Iacob from the Russian-language ed.). Basel: Birkhäuser Verlag. pp. viii+156. ISBN 3-7643-5401-1. MR 1442255.
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