Geodesic
On some functions of a linear closed operator
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 173-187 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A densely defined linear injective operator whose regular set contains the negative real semiaxis is given in a complex Banach space. A power asymptotic bound is known for the norm of the resolvent of this operator at zero and at infinity on the same semiaxis. We study some classes of functions of this operator constructed (in view of the Cauchy integral formula) on the basis of the corresponding scalar analytic functions that have power asymptotic bounds for their moduli at zero and at infinity. A number of properties of the operator functions are established, in particular, the multiplicative property and the invertibility. It is proved that a linear combination of integer powers of an arbitrary linear injective operator with nonempty resolvent set (under natural constraints on the coefficients) is a closed operator and that functions of an operator $A$ constructed for a linear combination of scalar power functions with integer exponents coincide with the corresponding linear combination of powers of this operator.
Keywords: linear closed operator, functions of an operator, multiplicative property, invertibility. 
@article{TIMM_2016_22_4_a16,
     author = {L. F. Korkina and M. A. Rekant},
     title = {On some functions of a linear closed operator},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {173--187},
     year = {2016},
     volume = {22},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a16/}
}
TY  - JOUR
AU  - L. F. Korkina
AU  - M. A. Rekant
TI  - On some functions of a linear closed operator
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2016
SP  - 173
EP  - 187
VL  - 22
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a16/
LA  - ru
ID  - TIMM_2016_22_4_a16
ER  - 
%0 Journal Article
%A L. F. Korkina
%A M. A. Rekant
%T On some functions of a linear closed operator
%J Trudy Instituta matematiki i mehaniki
%D 2016
%P 173-187
%V 22
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a16/
%G ru
%F TIMM_2016_22_4_a16
L. F. Korkina; M. A. Rekant. On some functions of a linear closed operator. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 173-187. http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a16/

[1] Danford N., Shvarts Dzh.T., Lineinye operatory. Obschaya teoriya, Izd-vo inostr. lit., M., 1962, 895 pp.

[2] Rudin U., Funktsionalnyi analiz, Mir, M., 1975, 449 pp. | MR

[3] Krein S.G., Lineinye differentsialnye uravneniya v banakhovykh prostranstvakh, Nauka, M., 1967, 275 pp. | MR

[4] M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik, P.E. Sobolevskii, Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966, 499 pp. | MR

[5] Sobolevskii P.E., Chebotareva L.M., “O drobnykh stepenyakh plokho pozitivnykh operatorov”, Tr. mat. fak-ta Voronezh. un-ta, no. 3, Voronezh, 1971, 112–118

[6] Martinez C., Miguel S., Javier P., “A functional calculus and fractional powers for multivalued linear operators”, Osaka J. Math., 37:3 (2000), 551–576 | MR | Zbl

[7] Mirotin A.R., “O mnogomernom funktsionalnom ischislenii Bokhnera - Fillipsa”, Problemy fiziki, matematiki i tekhniki, 1:1 (2009), 60–63 | MR

[8] Korkina L.F., Rekant M.A., “Nekotorye klassy funktsii lineinogo zamknutogo operatora”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17:3 (2011), 186–200

[9] Korkina L.F., Rekant M.A., “Drobnye stepeni odnogo klassa operatorov”, Izv. vuzov. Matematika, 1991, no. 9, 81–83 | MR

[10] Korkina L.F., Rekant M.A., “Rasshirenie klassa stepennykh operatornykh funktsii”, Matematika i mekhanika, v. 8, Izv. Ural. gos. un-ta, no. 38, 2005, 80–90 | MR | Zbl