About operator functions of an operator variable

A family of operator functions for which the domain and the range of values are included in the real Banach algebra of bounded linear operators acting in a real Banach space is considered. Such functions find application in the study of linear differential equations in a Banach space. Known operator functions are studied: exponential, sine, cosine, hyperbolic sine, hyperbolic cosine determined by the sums of the corresponding operator power series. For the functions of sine, cosine, hyperbolic sine, hyperbolic cosine, addition formulas are indicated, from which there follow the formulas for transforming the product of operator trigonometric functions and operator hyperbolic functions into a sum as well as those for transforming the sum and difference of operator trigonometric functions of the same name and operator hyperbolic functions of the same name into a product. The basic operator hyperbolic identity is proved. The concepts of the following operator functions are introduced: tangent, cotangent, secant, cosecant, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, hyperbolic cosecant. The periodicity of operator trigonometric functions of sine, cosine, tangent, cotangent, and the reduction formulas for them are proved. Relationships between operator functions of tangent and cotangent, hyperbolic tangent and hyperbolic cotangent are found. One useful application of the obtained operator trigonometric formulas is pointed out: it is proved that the operator functions ${Y}_{1}(t) = \sin Bt,$ ${Y}_{2}(t) = \cos Bt$ are infinitely differentiable on $\mathbb{R};$ formulas for the derivatives of any order of these functions are found.