sup norm . Let be a continuous function with . The operator with domain is a closed densely defined operator and generates the multiplication semigroup where Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices and a lot of the properties of can be derived by properties of . For example is bounded on if and only if is bounded. Translation semigroup Let be the space of bounded, uniformly continuous functions on endowed with the sup norm. The (left) translation semigroup is given by . Its generator is the derivative with domain . Abstract Cauchy problems Consider the abstract Cauchy problem: where A is a closed operator on a Banach space X and x∈X. There are two concepts of solution of this problem: a continuously differentiable function u: [0, ∞) → X is called a classical solution of the Cauchy problem if u(t&hairsp;) ∈ D(A) for all t > 0 and it satisfies the initial value problem, a continuous function u: [0, ∞) → X is called a mild solution of the Cauchy problem if Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable. The following theorem connects abstract Cauchy problems and strongly continuous semigroups. Theorem: Let A be a closed operator on a Banach space X. The following assertions are equivalent: for all x∈X there exists a unique mild solution of the abstract Cauchy problem, the operator A generates a strongly continuous semigroup, the resolvent set of A is nonempty and for all x ∈ D(A) there exists a unique classical solution of the Cauchy problem. When these assertions hold, the solution of the Cauchy problem is given by u(t&hairsp;) = T(t&hairsp;)x with T the strongly continuous semigroup generated by A. Generation theorems In connection with Cauchy problems, usually a linear operator A is given and the question is whether this is the generator of a strongly continuous semigroup. Theorems which answer this question are called generation theorems. A complete characterization of operators that generate exponentially bounded strongly continuous semigroups is given by the Hille–Yosida theorem. Of more practical importance are however the much easier to verify conditions given by the Lumer–Phillips theorem. Special classes of semigroups Uniformly continuous semigroups The strongly continuous semigroup T is called uniformly continuous if the map t → T(t&hairsp;) is continuous from [0, ∞) to L(X).