About complex operator functions of a complex operator variable

We consider a family of complex operator functions whose domain and range of values are included in the real Banach algebra of bounded linear complex operators acting in the Banach space of complex vectors over the field of real numbers. It is shown that the study of a function from this family can be reduced to the study of a pair of real operator functions of two real operator variables. The main elementary functions of this family are considered: power function; exponent; trigonometric functions of sine, cosine, tangent, cotangent, secant, cosecant; hyperbolic sine, cosine, tangent, cotangent, secant, cosecant; the main property of the exponent is proved. A complex Euler operator formula is obtained. Relations that express sine and cosine in terms of the exponent are found. For the trigonometric functions of sine and cosine, addition formulas are justified. The periodicity of the exponent and trigonometric functions of sine, cosine, tangent, cotangent is proved; reduction formulas for these functions are provided. The main complex operator trigonometric identity is obtained. Equalities connecting trigonometric and hyperbolic functions are found. The main complex operator hyperbolic identity is established. For the hyperbolic functions of sine and cosine, addition formulas are indicated. As an example of an elementary function from the family of complex operator functions under consideration, a rational function is considered, a special case of which is the characteristic operator polynomial of a linear homogeneous differential equation of $n$-th order with constant bounded operator coefficients in a real Banach space.