Compactness and norm continuity of the difference of two cosine functions (Q1428959)

Given two exponentially bounded cosine functions \(C(t,A)\) and \(C(t,B)\) on a Banach space \(X\) generated by the operators \(A\) and \(B\), respectively, the compactness of the difference operator \(C(t,A)-C(t,B)\) is studied in terms of the difference of the resolvents \(R(\lambda,A)-R(\lambda,B)\). If \(C(t,A)-C(t,B)\) is assumed to be norm continuous for \(t>0\), then its compactness for \(t\geq0\) is equivalent to the compactness of \(R(\lambda,A)-R(\lambda,B)\) for all sufficiently large \(\lambda\). Further slightly refined results are also obtained and a similar problem is studied for the difference of sine functions \(S(t,A)-S(t,B)\). For bounded perturbations of cosine functions, the norm continuity is automatic, resulting in a description of the compactness of \(C(t,A)-C(t,A+B)\) in terms of resolvents. It is also shown that if \(B\) is bounded and \(C(t,A)-C(t,A+B)\) is compact for all \(t>0\), then \(A-B\) must be compact. Finally, the relationship between the compactness of \(C(t,A)-C(t,B)\) and that of \(e^{tA}-e^{tB}\) is investigated in the same spirit.