### SOME PROPERTIES OF OPERATOR EXPONENT

#### Abstract

We study operators given by series, in particular, operators of the form \(e^B=\sum\limits_{n=0}^{\infty}{B^n}/{n!},\) where \(B\) is an operator acting in a Banach space \(X\). A corresponding example is provided. In our future research, we will use these operators for introducing and studying functions of operators constructed (with the use of the Cauchy integral formula) on the basis of scalar functions and admitting a faster than power growth at infinity.

#### Keywords

Closed operator, Operator exponent, Multiplicative property

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- Balakrishnan A.V. Fractional powers of closed operators and the semigroups generated by them.
*Pacific J. Math.*, 1960. Vol. 10. No. 2. P. 419–437. URL: https://projecteuclid.org/euclid.pjm/1103038401 - Dunford N., Schwartz J.T.
*Linear Operators Part I: General Theory.*New York: Interscience Publishers, 1958. 858 p. - Korkina L.F., Rekant M.A. An extension of the class of power operator functions.
*Izvestiya Uralskogo gosudarstvennogo universiteta (Matematika i mekhanika)*[Bulletin of the Ural State University (Mathematics and Mechanics)], 2005. No. 38. P. 80–90. (in Russian) URL: http://hdl.handle.net/10995/24591 - Korkina L.F., Rekant M.A. Some classes of functions of a linear closed operator.
*Proc. Steklov Inst. Math.*, 2012. Vol. 277, Suppl. 1. P. 121–135. DOI: 10.1134/S0081543812050124 - Korkina L.F., Rekant M.A. Properties of mappings of scalar functions to operator functions of a linear closed operator.
*Trudy Inst. Mat. i Mekh. UrO RAN*[Proc. of Krasovskii Institute of Mathematics and Mechanics of the UB RAS], 2015. Vol. 21, No. 1. P. 153–165. (in Russian) URL: http://mi.mathnet.ru/eng/timm/v21/i1/p153 - Krein S.G.
*Lineinye differentsial’nye uravneniya v banakhovom prostranstve*[Linear Differential Equations in Banach Space]. Moscow: Nauka, 1967. 464 p. (in Russian) - Lusternik L.A., Sobolev V.J.
*Elements of functional analysis.*Delhi: Hindustan Publishing Corpn., 1974. 376 p. - Rudin W.
*Functional Analysis.*New York: McGraw-Hill, 1973. 397 p.

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