"Applied analysis" by Hunter defines the resolvent set as
The resolvent set $\rho(A)$ of an unbounded operator $A:D(A)\subset\mathcal{H} \to \mathcal{H}$ consists of complex numbers $\lambda$ such that $A-\lambda I$ is a one-to-one, onto map from $D(A)$ to $\mathcal{H}$, and $(A-\lambda I)^{-1}$ is bounded.
Here $\mathcal{H}$ is a separable Hilbert space.
"Introductory functional analysis with applications" by Kreyszig defines the resolvent set as
Let $X \ne \{0\}$ be a complex normed space and $A:D(A) \to X$ a linear operator with domain $D(A) \subset X$. A regular value $\lambda$ of $A$ is a complex number such that i) $R_\lambda(A)$ exists, ii) $R_\lambda(A)$ is bounded, iii) $R_\lambda(A)$ is defined on a set which is dense in $X$. The resolvent set $\rho(A)$ of $A$ is the set of all regular values of $A$.
Here $R_\lambda(A)$ is $(A - \lambda I)^{-1}$.
The first definition requires $R_\lambda(T)$ to be defined on the whole Hilbert space, but the second definition requires it to be defined on a dense subspace (although it is stated with normed space, I think the same definition is used for Hilbert spaces also). The selected answer to a similar question says that the two definitions are not equivalent. Are both of the two definitions commonly used? If so, does the difference not make any significant difference on the theorems of spectral theory?
