(Cross-post from Math Stackexchange, where some work has been done in the comments)

Let $T,K$ be unbounded operators on a Hilbert space $H$. I've seen the following definition of a relatively compact operator:

(i) The operator $K$ is called

relatively compactwith respect to $T$, if for some $z$ in the resolvent set of $T$, $KR_T(z)$ is compact, where $R_T(z):=(T-z)^{-1}.$

I've also seen:

(ii) The operator $K$ is called

relatively compactwith respect to $T$, if for every sequence $(x_n)_{n \in \mathbb{N}}\subseteq H$ such that $(Tx_n)_{n \in \mathbb{N}}$ is bounded, $(Kx_n)_{n \in \mathbb{N}}$ contains a convergent subsequence.

All of this is in the context of spectral theory and $T$ can be assumed to be self-adjoint.
Do definitions (i) and (ii) have something to do with each other, or are they distinct? *What is the intuition behind these definitions?* Definition (ii) looks like a generalisation of a compact operator, but definition (i) is just weird.