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About a complex operator exponential function of a complex operator argument main property

Annotation

Operator functions e^A, sin B, cos B of the operator argument from the Banach algebra of bounded linear operators acting from E to E are considered in the Banach space E . For trigonometric operator functions sin B, cos B, formulas for the sine and cosine of the sum of the arguments are derived that are similar to the scalar case. In the proof of these formulas, the composition of ranges with operator terms in the form of Cauchy is used. The basic operator trigonometric identity is given. For a complex operator exponential function e^Z of an operator argument Z from the Banach algebra of complex operators, using the formulas for the cosine and sine of the sum, the main property of the exponential function is proved. Operator functions e^At , sin Bt, cos Bt, e^Zt of a real argument t∈(-∞;∞) are considered. The facts stated for the operator functions of the operator argument are transferred to these functions. In particular, the group property of the operator exponent e^Zt is given. The rule of differentiation of the function e^Zt is indicated. It is noted that the operator functions of the real argument t listed above are used in constructing a general solution of a linear n th order differential equation with constant bounded operator coefficients in a Banach space.

Keywords

Banach space; Banach algebra; exponential operator function; trigonometric operator functions; exponential operator function main property; the composition of operator ranges in the form of Cauchy; basic operator trigonometric identity

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DOI

10.20310/2686-9667-2019-24-127-324-332

UDC

517.937

Pages

324-332

References

[1] V. I. Fomin, “On the general solution of a linear n th-order differential equation with constant bounded operator coefficients in a Banach space”, Differential Equations, 41:5 (2005), 687–692. [2] V. I. Fomin, “On the case of multiple roots of the characteristic operator polynomial of an n th-order linear homogeneous differential equation in a Banach space”, Differential Equations, 43:5 (2007), 732–735. [3] V. I. Fomin, “About a solutions family of a linear homogeneous differential equation of the n thorder in a Banach space”, Actual Areas of Research of the 21th Century: Theory and Practice, 6:42 (2018), 382–384 (In Russian). [4] V. I. Fomin, “About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators”, Tambov University Reports. Series: Natural and Technical Sciences, 24:126 (2019), 237–243 (In Russian). [5] V. A. Trenogin, Functional Analysis, Nauka, Moscow, 1980 (In Russian). [6] Y. L. Daleckiy , M. G. Kreyn, Stability of Solutions of Differential Equations in a Banach Space, Nauka, Moscow, 1970 (In Russian). [7] V. I. Fomin, “About a complex operator Banach algebra”, Tambov University Reports. Series: Natural and Technical Sciences, 23:124 (2018), 813–823, DOI: 10.20310/1810-0198-2018-23-124-813-823 (In Russian). [8] V. I. Fomin, “About the main operator trigonometric identity”, Modern Methods of the Theory of Boundary Value Problems, Voronezh Spring Mathematical School «Pontryagin Readings – XXX» (Voronezh, May 3–9, 2019), Materials of the International Conference, VSU Publishing House, Voronezh, 2019, 284–285 (In Russian).

Received

2019-05-15

Section of issue

Scientific articles

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