On the compactness of fractional resolvent operator functions (Q343478)

Let \(X\) be a complex Banach space and \(A\) be a closed linear operator on \(X\). Let \(0<\alpha\leq1\) and \(\{S_\alpha(t)\}_{t\geq0}\) be an \((\alpha,1)\)-resolvent family \(S_\alpha(t)\) of type \((M,\omega)\) generated by \(A\). Assume that \(S_\alpha(t)\) is uniformly continuous for \(t>0\). Then \(S_\alpha(t)\) is a compact operator for all \(t>0\) iff \((\mu-A)^{-1}\) is a compact operator for all \(\mu>\omega^{1/\alpha}\). Let \(1<\alpha\leq 2\) and \(A\) be the generator of an \((\alpha,1)\)-resolvent family \(S_\alpha(t)\) of type \((M,\omega)\). Then \(A\) generates an \((\alpha,\alpha)\)-resolvent family \(R_\alpha(t)\) of type \((M/(\alpha-1),\omega)\). Moreover, \(R_\alpha(t)\) is a compact operator for all \(t>0\) iff \((\mu-A)^{-1}\) is a compact operator for all \(\mu>\omega^{1/\alpha}\). Some applications to a semilinear problem with non-local initial condition are given.