Geodesic
On the product of operator functions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 203-212 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In a Banach space, a linear densely defined operator $A$ and some closed domain $\overline{G}$ lying in the regular set of $A$ and containing the nonpositive real semiaxis are given. A power estimate for the norm of the resolvent of $A$ in the domain $G$ is assumed to be known. Under the assumption that the operators $e^{uA}$ defined by power operator series are closed for $u>0$, two classes of functions of this operator are introduced and studied. The construction of these classes is based on the integral Cauchy formula with corresponding scalar functions analytic in the complement of $G$ and such that their modules have an exponential estimate in the complement of $G$. If the operator $A$ satisfies certain constraints, then the introduced classes of functions of $A$ are extensions of the corresponding classes of operator functions, which we studied earlier jointly with L. F. Korkina. The multiplicative property of the operator functions is established, and the question of their invertibility is considered.
Keywords: functions of an operator, operator exponent, multiplicative property. 
@article{TIMM_2024_30_1_a14,
     author = {M. A. Rekant},
     title = {On the product of operator functions},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {203--212},
     year = {2024},
     volume = {30},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a14/}
}
TY  - JOUR
AU  - M. A. Rekant
TI  - On the product of operator functions
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2024
SP  - 203
EP  - 212
VL  - 30
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a14/
LA  - ru
ID  - TIMM_2024_30_1_a14
ER  - 
%0 Journal Article
%A M. A. Rekant
%T On the product of operator functions
%J Trudy Instituta matematiki i mehaniki
%D 2024
%P 203-212
%V 30
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a14/
%G ru
%F TIMM_2024_30_1_a14
M. A. Rekant. On the product of operator functions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 203-212. http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a14/

[1] Danford N., Shvarts Dzh.T., Lineinye operatory. Obschaya teoriya, Iz-vo inostr. lit., M., 1962, 896 pp.

[2] Lyusternik L.A., Sobolev S.L., Elementy funktsionalnogo analiza, Nauka, M., 1965, 519 pp. | MR

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

[4] Balakrishnan A.V., “Fractional powers of closed operators and semigroups generated by them”, Pacific J. Math. Soc., 3 (1960), 419–437 | DOI | MR

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

[6] Krein S.G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967, 494 pp.

[7] Komatsu H., “Fractional powers of operators. Interpolation spaces”, Pacific J. Math., 21:1 (1967), 89–111 | DOI | MR | Zbl

[8] Kostin V.A., Kostin D.V., Kostin A.V., “Operatornye kosinus-funktsii i granichnye zadachi”, Dokl. AN, 486:5 (2019), 531–536 | DOI

[9] Abdullaev O.Kh., “Nelokalnaya zadacha s integralnym usloviem skleivaniya dlya nagruzhennogo parabolo-giperbolicheskogo uravneniya s drobnoi proizvodnoi Kaputo”, Differents. uravneniya, 59:3 (2023), 350–357 | DOI | Zbl

[10] Korkina L.F., Rekant M.A., “Svoistva otobrazhenii skalyarnykh funktsii v operatornye lineinogo zamknutogo operatora”, Tr. Instituta matematiki i mekhaniki UrO RAN, 21:1 (2015), 153–165 | MR

[11] Korkina L.F., Rekant M.A., “O proizvedenii operatornykh eksponent”, Tr. Instituta matematiki i mekhaniki UrO RAN, 28:1 (2022), 156–163 | DOI | MR