Functionals-preserving cosine families generated by Laplace operators in C[0,1]
Adam Bobrowski, Adam Gregosiewicz, Małgorzata Murat · Discrete and Continuous Dynamical Systems - B · 2015
Let \( C[0,1] \) be the space of continuous functions on the unit interval \([0,1] \).A cosine family $\{C(t), t \in \mathbb{R}\}$ in $C[0,1]$ is said to be Laplace-operatorgenerated, if its generator is a restriction of the Laplace operator $L\colon f\mapsto f''$ to a suitable subset of $C^2[0,1].$ The family is said to preservea functional $F \in (C[0,1])^*$ if for all $f \in C[0,1]$ and $t \in \mathbb{R}, $$FC(t)f = Ff.$ We study a class of pairs of functionals such that for eachmember of this class there is a unique Laplace-operator generated cosine familythat preserves both functionals in the pair.