Operator representation of sectorial linear relations and applications

Let ℋ be a Hilbert space with inner product 〈⋅,⋅〉 and let T be a non-densely defined linear relation in ℋ with domain D(T). We prove that if T is sectorial then it can be expressed in terms of some sectorial operator A with domain D(A)=D(T) and that T is maximal sectorial if and only if A is maximal sectorial in D(T)¯. The operator A has the property that for every u∈D(A) and every v∈D(T) and any u′∈T(u), 〈Au,v〉=〈u′,v〉. We use this representation to show that every sectorial linear relation T is form closable, meaning that the form associated with T has a closed extension. We also prove a result similar to Kato’s first representation theorem for sectorial linear relations. Unlike the results available in the literature, we do not assume that the graph of the linear relation T is a closed subspace of H×H.