Evolution equations

Let X,Y be normed spaces. The set of bounded linear operators is noted as L(X,Y ). Let now D = D(A) ⊂ X be a linear subspace, and A : D −→ Y a linear (not necessarily bounded!) operator. Notation: (A,D(A)) : X −→ Y Definition: G(A) := {(x,Ax) |x ∈ D} is called the graph of A. Obviously, G(A) is a linear subspace of X × Y . The linear operator A is called closed if G(A) is closed in X × Y . The linear operator A is called closable if G(A) = G(A) for some linear operator (A,D(A)) : X −→ Y . In this case A is called the closure of A. If a closure exists, it is obviously unique. Clearly, if A is closed then A is closable and A = A.