Uniform continuity and compactness for resolvent families of operators
type: Item
instance of: scholarly article
title: Uniform continuity and compactness for resolvent families of operators
zbMATH Open document ID: 0842.45009
DOI: 10.1007/BF00992842
author: Carlos Lizama
published in: Acta Applicandae Mathematicae
publication date: 1996-07-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 47D09
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 790354
zbMATH Keywords: semigroup of operators
zbMATH Keywords: Volterra convolution equation
zbMATH Keywords: Banach space
zbMATH Keywords: resolvent family
zbMATH Keywords: uniform continuity
zbMATH Keywords: compactness
zbMATH Keywords: cosine-families of operators
reviewed by: Roland Duduchava
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 790354