M.A. Rekant. On the product of operator functions ... P. 203-212

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

Received September 4, 2023

Revised November 7, 2023

Accepted November 13, 2023

Mark Aleksandrovich Rekant, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: M.A.Rekant@urfu.ru

REFERENCES

1.   Dunfrod N., Schwartz J. Linear operators: General theory, New York, London, Interscience publishers, 1958. ISBN: 9780470226056. Translated to Russian under the title Lineinye operatory: Obshchaya teoriya, Moscow, Inostr. Liter. publ., 1962, 895 p.

2.   Lusternik L.A., Sobolev V.J. Elements of functional analysis, International monographs on advanced mathematics and physics, Delhi, Hindustan Publishing 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, 520 p.

3.   Rudin W. Functional Analysis. N.-Y., McGraw–Hill, 1973, 397 p. ISBN: 9780070542259. 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. 3, 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 operators in spaces of summable functions. Netherlands, Springer, 1976, 536 p. ISBN: 978-94-010-1544-8. Original Russian text published in Krasnosel’skii M.A., Zabreiko P.P., Pustyl’nik E.I., Sobolevskii P.E. Integral’nye operatory v prostranstvakh summiruemykh funktsii, Moscow, Nauka Publ., 1966, 499 p.

6.   Krein S. Linear differential equations in Banach space, Translations of Mathematical Monographs, vol. 29. Providence, Amer. Math. Soc., 1972, 390 p. ISBN: 978-1-4704-1628-7. Original Russian text published in Krein S.G., Lineinye differentsial’nye uravneniya v banakhovom prostranstve, Moscow, Nauka Publ., 1967, 494 p.

7.   Komabsu 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.   Kostin V.A., Kostin D.V., Kostin A.V. Operator cosine functions and boundary value problems. Doklady Math., 2019, vol. 99, no. 3, pp. 303–307. doi: 10.1134/S1064562419030177

9.   Abdullaev O.Kh. A nonlocal problem with an integral matching condition for a loaded parabolic-hyperbolic equation with a fractional Caputo derivative. Diff. Equ., 2023, vol. 59, no. 3, pp. 351–358. doi: 10.1134/S0012266123030059

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. On the product of operator exponentials. Tr. Inst. Math. Mekh. UrO RAN, 2022, vol. 28, no. 1, pp. 156–163 (in Russian). doi: 10.21538/0134-4889-2022-28-1-156-163

Cite this article as: M.A. Rekant. On the product of operator functions. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 203–212.