First representation theorem for sesquilinear forms - what is the role of the "core"?

Question

In the first representation theorem, the notion of the core of a sesquilinear form appears. What is the intuition behind this notion, in context of this theorem and in general? I appreciate any comments and answers!

(Here are some relevant definitions and background. All excerpts are from Kato's Perturbation Theory for Linear Operators, pp.308-322.)

Answer 1

The definition of core is given in the same Kato's book that you cited at page 317 for a closed sectorial form and at page 166 for a closed operator.

I refer to this last (but the first is a particular case).

If $T: X \rightarrow Y$ is a closed operator and $\mathbf{D}(T)$ its domain, than a subspace $\mathbf{D}$ of $\mathbf{D}(T)$ is a core of $T$ if the set $\mathbf{G}=\{ (u,Tu) : u \in \mathbf{D} \}$ is such its closure is the graph $\mathbf{G}(T)$ of $T$.

Note that the closure of $\mathbf{G}$ is in the induced topology on $X\times Y$.

As an intuition i think that we can see a core $\mathbf{D}$ as a subspace of the domain of $T$ such that if we take a restriction of the operator $T$ to this subspace we can reconstruct its graph by the values on $\mathbf{D}$ taking limits of convergent sequences $(u_n,Tu_n)$ for $u_n \in \mathbf{D}$.