I'm currently working with partial isometries over $B(H)$ (the set of all bounded linear operators over some Hilbert space $H$) whose range is $\infty$-dim, and naturally require examples to test my thoughts and results.
I'm mostly interested in operators of this type which are not projections or isometries (over all of $H$). So far, the only examples I've come up with which do not fall under these two subclasses are based on (powers of) the left and right-shift operators on $\ell_2$.
Definition
A partial isometry is an operator $T \in B(H)$ such that $\lVert Tx \rVert = \lVert x \rVert$ for all $x \in \ker{T}^\perp$ (for a regular isometry, $\ker{T} = \{0\}$, so the norm of all elements of $H$ is preserved).
Whenever an operator is a partial isometry, its adjoint also is. Additionally, $T$ being a partial isometry is equivalent to $TT^*$ being a projection (same goes for $T^*T$) and it is also equivalent to $T^*TT^* = T^*$ (or $TT^*T=T$) being true.
Let $T$ be a partial isometry of $H$, with kernel $K^\perp$ and range $L$.
The restriction $U$ of $T$ to $K$ is then an isometry onto $L$, and $T=U\circ P$, where $P$ is the orthogonal projection of $H$ onto $K$. Conversely, given two isometric closed subspaces $L,K\subseteq H$, such a composition $T:=U\circ P$ is a partial isometry, with range $L$.
Equivalently, every partial isometry $T:H\to H$ is given by $$T(e_i)=\begin{cases}f_i&\text{if }i\in I\\0&\text{if }i\in J\end{cases}$$ where $(e_i)_{i\in I\sqcup J},(f_i)_{i\in I\sqcup K}$ are two Hilbert basis of $H$.