A linear densely defined operator $A$ and a domain lying in its regular set and containing the nonpositive real semiaxis are given in a Banach space. A power bound for the norm of the resolvent of the operator at infinity is assumed to be known. The operators $e^{tA}$ $(t\in \mathbb R)$, given by the corresponding series, and $(e^{tA})_{I}$ for $t<0$, introduced on the basis of the integral Cauchy formula, are considered. The question of invertibility of the operator exponentials and the multiplicative property of these exponentials are studied. The operator exponentials can be used for the construction of operator functions of a wider class than that considered by the authors earlier.
Keywords: operator exponent, operator functions, multiplicative property
Received October 22, 2021
Revised November 30, 2021
Accepted December 12, 2021
Lyudmila Fedorovna Korkina, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: L.F.Korkina@urfu.ru
Mark Aleksandrovich Rekant, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: M.A.Rekant@urfu.ru
REFERENCES
1. Dunford N., Schwartz J.T. Linear operators. I. General theory. NY: Interscience Publ., 1958, 858 p. ISBN: 0470226056 . Translated to Russian under the title Lineinye operatory. Obshchaya teoriya. Moscow: Inostr. Lit. Publ., 1962, 896 p.
2. Lusternik L.A. Sobolev V.I. Elements of functional analysis. Delhi: Hindustan Publ. Corp., 1974, 360 p. ISBN: 0470556501 . Original Russian text published in Lyusternik L.A., Sobolev V.I. Elementy funktsional’nogo analiza, Moscow: Nauka Publ., 1965, 519 p.
3. Rudin W. Functional analysis. NY: McGraw-Hill, 1973, 397 p. ISBN: 978-0070542259 . Translated to Russian under the title Funktsional’nyi analiz. Moscow: Mir Publ., 1975, 449 p.
4. Balakrishnan A.V. Fractional powers of closed operators and semigroups generated by them. Pacific J. Math. Soc., 1960, vol. 10, no. 2, pp. 419–437. doi: 10.2140/pjm.1960.10.419
5. Krasnosel’skii M.A., Zabreiko P.P., Pustyl’nik E.I., Sobolevskii P.E. Integral’nye operatory v prostranstvakh summiruemykh funktsii [Integral operators in the spaces of summable functions]. Moscow: Nauka Publ., 1966, 499 p.
6. Krein S.G. Linear differential equations in Banach space. Providence: AMS, 1972, 390 p. doi: 10.1090/mmono/029 . Original Russian text published in Krein S.G. Lineinye differentsial’nye uravneniya v banakhovom prostranstve, Moscow: Nauka Publ., 1967, 494 p.
7. Komatsu H. Fractional powers of operators. II. Interpolation spaces. Pacific J. Math., 1967, vol. 21, no. 1, pp. 89–111. doi: 10.2140/pjm.1967.21.89
8. Repin O.A. On a problem for a mixed-type equation with fractional derivative. Russ. Math., 2018, vol. 62, no. 8, pp. 38–42. doi: 10.3103/S1066369X18080066
9. Kostin V.A., Kostin D.V., Kostin A.V. Operator cosine functions and boundary value problems. Dokl. Math., 2019, vol. 99, no. 3, pp. 303–307. doi: 10.1134/S1064562419030177
10. Korkina L.F., Rekant M.A. Properties of mappings of scalar functions to operator functions of a linear closed operator. Tr. Inst. Math. Mekh. UrO RAN, 2015, vol. 21, no. 1, pp. 153–165 (in Russian).
11. Korkina L.F., Rekant M.A. Certain properties of operator exponent. Ural. Math. J., 2018, vol. 4, no. 2, pp. 33–42. doi: 10.15826/umj.2018.2.005
Cite this article as: L.F. Korkina, M.A. Rekant. On the product of operator exponentials, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 156–163.
