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

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 $t \in ( - \infty ,\infty )$ 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.