摘要:
Using the operator functional relation Sh(t+s)+Sh(t−s) = 2[I +2Sh2(t2)] Sh(s), Sh(0) = 0, we introduce and study the strongly continuous sine function Sh(t), t ∈ (−∞, ∞), of linear bounded transformations acting in a complex Banach space E. Also, we study the cosine function Ch(t) given by the equation Ch(t) = I + 2Sh2(t2), where I is the identity operator in E. The pair Ch(t) and Sh(t) is called an exponential trigonometric pair (ETP, in brief). For such pairs, we determine the generating operator (generator) by the equation Sh″ (0)φ = Ch″ (0)φ = Aφ and we give a criterion for A to be the generator of ETP. We find a connection between Sh(t) and the uniform well-posedness of the Cauchy problem with Krein’s condition for the equation d2u(t)dt2 = Au(t). This problem is uniformly well-posed if and only if A is the exponent generator of the sine function Sh(t). We introduce the concept of a bundle of several ETPs, which also forms an ETP, and we give a representation for bundle’s generator. The facts obtained significantly expand the applicability of operator methods to study the well-posedness of initial-boundary value problems. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.