Integral representations of unbounded operators by infinitely smooth kernels

Igor Novitskiî

Open Mathematics (2005)

  • Volume: 3, Issue: 4, page 654-665
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Igor Novitskiî. "Integral representations of unbounded operators by infinitely smooth kernels." Open Mathematics 3.4 (2005): 654-665. <http://eudml.org/doc/268926>.

@article{IgorNovitskiî2005,
abstract = {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.},
author = {Igor Novitskiî},
journal = {Open Mathematics},
keywords = {47G10; 45P05},
language = {eng},
number = {4},
pages = {654-665},
title = {Integral representations of unbounded operators by infinitely smooth kernels},
url = {http://eudml.org/doc/268926},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Igor Novitskiî
TI - Integral representations of unbounded operators by infinitely smooth kernels
JO - Open Mathematics
PY - 2005
VL - 3
IS - 4
SP - 654
EP - 665
AB - 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.
LA - eng
KW - 47G10; 45P05
UR - http://eudml.org/doc/268926
ER -

References

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  1. [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. Zbl0767.42009
  2. [2] I.Ts. Gohberg and M.G. Kreîn: Introduction to the theory of linear non-selfadjoint operators in Hilbert space, Nauka, Moscow, 1965. 
  3. [3] E. Hernández and G. Weiss: A first course on wavelets, CRC Press, New York, 1996. 
  4. [4] P. Halmos and V. Sunder: Bounded integral operators on L 2spaces, Springer, Berlin, 1978. 
  5. [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. Zbl0208.16901
  6. [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. [7] V.B. Korotkov: Integral operators, Nauka, Novosibirsk, 1983. 
  8. [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. Zbl0911.45011
  9. [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. [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. [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. [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 Zbl0935.47024
  13. [13] I.M. Novitskiî: “Fredholm minors for completely continuous operators”, Dal’nevost. Mat. Sb., Vol. 7, (1999), pp. 103–122. 
  14. [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. [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. [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. [17] I.M. Novitskiî: Simultaneous unitary equivalence to Carleman operators with arbitrarily smooth kernels, arXiv:math.SP/0404274, April 15, 2004. 
  18. [18] J. Weidmann: “Carlemanoperatoren”, Manuscripta Math., Vol. 2 (1970), pp. 1–38. http://dx.doi.org/10.1007/BF01168477 

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