Operator sine-functions and trigonometric exponential pairs

With the help of operator functional relations $Sh(t+s)+Sh(t-s) = 2[ I+2 Sh^2(\frac t2)] Sh(s), Sh(0)=0,$ we introduce and study strongly continuous sine-function $Sh(t), t\in(-\infty, \infty),$ of linear bounded transformations acting in a complex Banach space $E$, together with the cosine-function $Ch(t)$ given by the equation $Ch(t)=I+2Sh^2(\frac t2)$, where $I$ is the identity operator in $E$.
The pair $Ch(t)$, $ Sh(t)$ is the exponential of a trigonometric pair (ETP). For such pairs a generating operator (generator) is determined by the equation $Sh''(0)\varphi = Ch''(0) \varphi = A \varphi$, and a criterion for $A$ to be the generator of the ETP is provided.
A relationship of $Sh(t)$ with the uniform well-posedness of the Cauchy problem with the Krein condition for the equation $\frac{d^2 u(t)}{dt^2}=Au(t)$ is described. This problem is uniformly well-posed if and only if $A$ is an exponent generator of the sine-function $Sh(t)$.
The concept of bundles of several ETP, which also forms a ETP, is introduced, and a representation for its generator is given.
The obtained facts expand significantly the possibilities of operator methods in the study of well-posed initial boundary value problems.