Uniform continuity and compactness for resolvent families of operators
The author studies the following Volterra convolution equation \[ V_A u(t):= u(t)- \int^t_0 k(t- s)Au (s) ds= f(t), \qquad t\in J:= [0,T ] \] where \(X\) is a Banach space, \(A\) a densely defined closed operator on \(X\), and \(k\in L^1_{\ell x} (\mathbb{R}_+)\). A strongly continuous family \(\{R(t) \}_{t\geq 0}\) of bounded operators on \(X\), commuting with \(A\), is called a resolvent family for the above equation if \(V_A Rx= x\) for all \(x\) in the domain of definition \({\mathcal D} (A)\) of \(A\). Conditions for the existence of \(R(T)\), their regularity, positivity and other properties of \(R(t)\) are widely studied. In the present work the uniform continuity of \(\{R(t) \}_{t\geq 0}\) and compactness of \(R(t)- I\) for \(t>0\) are characterized in terms of operator \(A\) provided the kernel \(k\) is regular. In particular cases \(k(t) \equiv 1\), \(k(t) \equiv t\) these results coincide with the known ones on \(C_0\)-semigroups and cosine-families of operators, obtained by J. Guthbert, H. Henriques, D. Lutz, J. Pruss and others.
scientific article; zbMATH DE number 790354