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.
@article{108062,
author = {S. M. Bahri},
title = {Spectral properties of a certain class of Carleman operators},
journal = {Archivum Mathematicum},
volume = {043},
year = {2007},
pages = {163-175},
zbl = {1164.05037},
mrnumber = {2354805},
language = {en},
url = {http://dml.mathdoc.fr/item/108062}
}
Bahri, S. M. Spectral properties of a certain class of Carleman operators. Archivum Mathematicum, Tome 043 (2007) pp. 163-175. http://gdmltest.u-ga.fr/item/108062/
Theory of Linear Operators in Hilbert Space, Dover, New York (1993). (1993) | MR 1255973 | Zbl 0874.47001
On the generalized resolvents of symmetric operators, Izv. Vizov Math. N${{}^\circ }$ 7, 1970. (1970)
On the extension of a certain class of Carleman operators, EMS, Z. Anal. Anwend. 26 (2007), 57–64. | MR 2306105 | Zbl 1142.47027
Théorie spectrale, Nouvelle série N$^{\circ }$ 8/C, 1982. (1982) | MR 0719649 | Zbl 0532.47014
Sur les équations intégrales singulières à noyau réel et symétrique, Uppsala Almquwist Wiksells Boktryckeri, 1923. (1923)
Cours d’analye T3, Presses polytechniques et universitaires romandes, 1998. (1998) | MR 1637558
Carleman and Korotkov operators on Banach spaces, Acta Sci. Math. 43 (1981), 111–119. (1981) | MR 0621370 | Zbl 0483.47033
Bounded integral operators on $L_{2}$-spaces, Ergeb. Math. Grenzgeb. 96, Springer 1978. (1978) | MR 0517709 | Zbl 0389.47001
On characteristic properties of Carleman operators, Sib. Math. J. 11, N$^\circ $1 (1970), 103-127. (1970) | MR 0283631
On Hermitian operators with deficiency indices one, Dokl. Akad. Nauk SSSR 43 (1944), 339–342. (Russian) (1944) | MR 0011170
Resolvents of a Hermitian operator with defect index $(m,m)$, Dokl. Akad. Nauk SSSR 52 (1946), 657–660. (Russian) (1946) | MR 0018341
Some new results in the theory of resolvents of Hermitian operators, Sov. Math. Dokl. 7 (1966), 1086–1089. (1966)
On a formula of the generalized resolvents of a nondensely defined Hermitian operator, Ukrain. Math. J. 44 (1992), 1522–1547. (1992) | MR 1215039
Methods of Hilbert spaces, PWN, 1967. (1967) | MR 0223910 | Zbl 0166.10203
On spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR 7 (1943), 285–296. (Russian) (1943) | MR 0010790 | Zbl 0061.26005
Allgemeine Eigenwerttheorie hermitescher Funktionaloperatoren, Math. Ann. 102 (1929–30), 49–131. (1929) | MR 1512569
Methods of Modern Mathematical Physics IV, Analysis of Operators, Academic Press, New York, 1978. (1978) | MR 0493421 | Zbl 0401.47001
Generalized Carleman operators, Indag. Math. 42 (1980), 49–59. (1980) | MR 0565943 | Zbl 0431.47028
On the theory of the resolvents of a symmetric operator with infinite deficiency indices, Dokl. Akad. Nauk Arm. SSR 44 (1965), 193–198. (Russian) (1965) | MR 0196497
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) (1955) | MR 0074642
Extensions and generalized resolvents of a non-densely defined symmetric operator, Math. USSR Izv. 4 (1970), 179–208. (1970) | MR 0278098
On Carleman integral operators, Proc. Amer. Math. Soc. 18, N$^\circ $3 (1967). (1967) | MR 0215139 | Zbl 0147.12002
Carleman Operators, Manuscripts, Math. N$^\circ $2 (1970), 1–38. (1970)
Linear Operators in Hilbert Spaces, Spring Verlag, New-York Heidelberg Berlin, 1980. (1980) | MR 0566954