Spectral properties of a certain class of Carleman operators

Bahri, S. M.

Archivum mathematicum, Tome 43 (2007) no. 3, pp. 163-175

Abstract (VO)

The object of the present work is to construct all the generalized spectral functions of a certain class of Carleman operators in the Hilbert space $L^{2}\left( X,\mu \right) $ and establish the corresponding expansion theorems, when the deficiency indices are (1,1). This is done by constructing the generalized resolvents of $A$ and then using the Stieltjes inversion formula.

Détail

Classification : 05C38, 15A15
Keywords: spectral theory; integral operator; defect indices

Bibliographie

[1] Akhiezer N. I., Glazman I. M.: Theory of Linear Operators in Hilbert Space. Dover, New York (1993). | MR | Zbl

[2] Alexandrov E. L.: On the generalized resolvents of symmetric operators. Izv. Vizov Math. N${{}^\circ }$ 7, 1970.

[3] Bahri S. M.: On the extension of a certain class of Carleman operators. EMS, Z. Anal. Anwend. 26 (2007), 57–64. | MR | Zbl

[4] Buchwalter H., Tarral D.: Théorie spectrale. Nouvelle série N$^{\circ }$ 8/C, 1982. | MR | Zbl

[5] Carleman T.: Sur les équations intégrales singulières à noyau réel et symétrique. Uppsala Almquwist Wiksells Boktryckeri, 1923.

[6] Chatterji S. D.: Cours d’analye T3. Presses polytechniques et universitaires romandes, 1998. | MR

[7] Gretsky N., Uhl J. J.: Carleman and Korotkov operators on Banach spaces. Acta Sci. Math. 43 (1981), 111–119. | MR | Zbl

[8] Halmos P. R., Sunder V. S.: Bounded integral operators on $L_{2}$-spaces. Ergeb. Math. Grenzgeb. 96, Springer 1978. | MR | Zbl

[9] Korotkov V. B.: On characteristic properties of Carleman operators. Sib. Math. J. 11, N$^\circ $1 (1970), 103-127. | MR

[10] Krein M. G.: On Hermitian operators with deficiency indices one. Dokl. Akad. Nauk SSSR 43 (1944), 339–342. (Russian) | MR

[11] Krein M. G.: Resolvents of a Hermitian operator with defect index $(m,m)$. Dokl. Akad. Nauk SSSR 52 (1946), 657–660. (Russian) | MR

[12] Krein M. G., Saakjan S. N.: Some new results in the theory of resolvents of Hermitian operators. Sov. Math. Dokl. 7 (1966), 1086–1089.

[13] Malamud M. M.: On a formula of the generalized resolvents of a nondensely defined Hermitian operator. Ukrain. Math. J. 44 (1992), 1522–1547. | MR

[14] Maurin K.: Methods of Hilbert spaces. PWN, 1967. | MR | Zbl

[15] Naimark M. A.: On spectral functions of a symmetric operator. Izv. Akad. Nauk SSSR 7 (1943), 285–296. (Russian) | MR | Zbl

[16] von Neumann J.: Allgemeine Eigenwerttheorie hermitescher Funktionaloperatoren. Math. Ann. 102 (1929–30), 49–131. | MR

[17] Reed M., Simon B.: Methods of Modern Mathematical Physics IV, Analysis of Operators. Academic Press, New York, 1978. | MR | Zbl

[18] Schep A. R.: Generalized Carleman operators. Indag. Math. 42 (1980), 49–59. | MR | Zbl

[19] Saakjan, Sh. N.: On the theory of the resolvents of a symmetric operator with infinite deficiency indices. Dokl. Akad. Nauk Arm. SSR 44 (1965), 193–198. (Russian) | MR

[20] Straus A. V.: Construction of the generalized spectral functions of a Sturm-Liouville differential operator on a half-axis. Izv. Akad. Nauk SSSR Ser. Mat. 19 (1955), 201–220. (Russian) | MR

[21] Straus A. V.: Extensions and generalized resolvents of a non-densely defined symmetric operator. Math. USSR Izv. 4 (1970), 179–208. | MR

[22] Targonski G. I.: On Carleman integral operators. Proc. Amer. Math. Soc. 18, N$^\circ $3 (1967). | MR | Zbl

[23] Weidman J.: Carleman Operators. Manuscripts, Math. N$^\circ $2 (1970), 1–38.

[24] Weidman J.: Linear Operators in Hilbert Spaces. Spring Verlag, New-York Heidelberg Berlin, 1980. | MR