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2005 | 3 | 4 | 654-665
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Integral representations of unbounded operators by infinitely smooth kernels

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EN
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EN
In this paper, we prove that every unbounded linear operator satisfying the Korotkov-Weidmann characterization is unitarily equivalent to an integral operator in L 2(R), with a bounded and infinitely smooth Carleman kernel. The established unitary equivalence is implemented by explicitly definable unitary operators.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
4
Strony
654-665
Opis fizyczny
Daty
wydano
2005-12-01
online
2005-12-01
Twórcy
Bibliografia
  • [1] P. Auscher, G. Weiss and M.V. Wickerhauser: “Local Sine and Cosine Bases of Coifman and Meyer and the Construction of Smooth Wavelets”, In: C.K. Chui (Ed.): Wavelets: a tutorial in theory and applications, Academic Press, Boston, 1992, pp. 237–256.
  • [2] I.Ts. Gohberg and M.G. Kreîn: Introduction to the theory of linear non-selfadjoint operators in Hilbert space, Nauka, Moscow, 1965.
  • [3] E. Hernández and G. Weiss: A first course on wavelets, CRC Press, New York, 1996.
  • [4] P. Halmos and V. Sunder: Bounded integral operators on L 2spaces, Springer, Berlin, 1978.
  • [5] V.B. Korotkov: “Classification and characteristic properties of Carleman operators”, Dokl. Akad. Nauk SSSR, Vol. 190(6), (1970), pp. 1274–1277; English transl.: Soviet Math. Dokl., Vol. 11(1), (1970), pp. 276–279.
  • [6] V.B. Korotkov: “Unitary equivalence of linear operators to bi-Carleman integral operators”, Mat. Zametki, Vol. 30(2), (1981), pp. 255–260; English transl.: Math. Notes, Vol. 30(1–2), (1981), pp. 615–617.
  • [7] V.B. Korotkov: Integral operators, Nauka, Novosibirsk, 1983.
  • [8] V.B. Korotkov: “Some unsolved problems of the theory of integral operators”, In: Sobolev spaces and related problems of analysis, Trudy Inst. Mat., Vol. 31, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1996, pp. 187–196; English transl.: Siberian Adv. Math., Vol. 7(2), (1997), pp. 5–17.
  • [9] I.M. Novitskiî: “Reduction of linear operators in L 2 to integral form with smooth kernels”, Dokl. Akad. Nauk SSSR, Vol. 318(5), (1991), pp. 1088–1091; English transl.: Soviet Math. Dokl., Vol. 43(3), (1991), pp. 874–877.
  • [10] I.M. Novitskiî: “Unitary equivalence between linear operators and integral operators with smooth kernels”, Differentsial’nye Uravneniya, Vol. 28(9), (1992), pp. 1608–1616; English transl.: Differential Equations, Vol. 28(9), (1992), pp. 1329–1337.
  • [11] I.M. Novitskii: “Integral representations of linear operators by smooth Carleman kernels of Mercer type”, Proc. Lond. Math. Soc. (3), Vol. 68(1), (1994), pp. 161–177.
  • [12] I.M. Novitskiî: “A note on integral representations of linear operators”, Integral Equations Operator Theory, Vol. 35(1), (1999), pp. 93–104. http://dx.doi.org/10.1007/BF01225530
  • [13] I.M. Novitskiî: “Fredholm minors for completely continuous operators”, Dal’nevost. Mat. Sb., Vol. 7, (1999), pp. 103–122.
  • [14] I.M. Novitskiî: “Fredholm formulae for kernels which are linear with respect to parameter”, Dal’nevost. Mat. Zh., Vol. 3(2), (2002), pp. 173–194.
  • [15] I.M. Novitskiî: Simultaneous unitary equivalence to bi-Carleman operators with arbitrarily smooth kernels of Mercer type, arXiv:math.SP/0404228, April 12, 2004.
  • [16] I.M. Novitskiî: Integral representations of closed operators as bi-Carleman operators with arbitrarily smooth kernels, arXiv:math.SP/0404244, April 13, 2004.
  • [17] I.M. Novitskiî: Simultaneous unitary equivalence to Carleman operators with arbitrarily smooth kernels, arXiv:math.SP/0404274, April 15, 2004.
  • [18] J. Weidmann: “Carlemanoperatoren”, Manuscripta Math., Vol. 2 (1970), pp. 1–38. http://dx.doi.org/10.1007/BF01168477
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475625
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