Integral representations of unbounded operators by infinitely smooth kernels

• Autorzy: Igor Novitskiî
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Closed linear operator; integral linear operator; Carleman kernel; characterization theorems for integral operators; 47G10; 45P05

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 4
• Strony: 654-665

Daty

wydano

2005-12-01

online

2005-12-01

Twórcy

autor

Igor Novitskiî

• Far-Eastern Branch of the Russian Academy of Sciences

Bibliografia

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• [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

Identyfikatory

DOI

10.2478/BF02475625