Existence Families, Functional Calculi and Evolution Equations

This monograph presents an operator-theoretic approach to ill-posed evolution equations. It presents the basic theory, and the more surprising examples, of generalizations of strongly continuous semigroups known as existent families and regularized semigroups. These families of operators may be used either to produce all initial data for which a solution in the original space exists, or to construct a maximal subspace on which the problem is well-posed. Regularized semigroups are also used to construct functional, or operational, calculi for unbounded operators.