Category:Resolvent Sets (Densely-Defined Linear Operators)

This category contains results about resolvent sets in the context of Densely-Defined Linear Operators.
Definitions specific to this category can be found in Definitions/Resolvent Sets (Densely-Defined Linear Operators).

Let $\HH$ be a Hilbert space over $\C$.

Let $\struct {\map D T, T}$ be a densely-defined linear operator.

We define the resolvent set of $T$, $\map \rho T$, as the set of $\lambda \in \C$ for which:

$T - \lambda I$ is injective, $\map {\paren {T - \lambda I} } {\map D T}$ is everywhere dense in $\HH$, and $\paren {T - \lambda I}^{-1}$ is bounded.