  Given Hilbert space operators $T, S\in\B$, let $\triangle$ and $\delta\in
B(\B)$ denote the elementary operators $\triangle_{T,S}(X)=(L_TR_S-I)(X)=TXS-X$
and $\delta_{T,S}(X)=(L_T-R_S)(X)=TX-XS$. Let $d=\triangle$ or $\delta$.
Assuming $T$ commutes with $S^*$, and choosing $X$ to be the positive operator
$S^{*n}S^n$ for some positive integer $n$, this paper exploits properties of
elementary operators to study the structure of $n$-quasi $[m,d]$-operators
$d^m_{T,S}(X)=0$ to bring together, and improve upon, extant results for a
number of classes of operators, amongst them $n$-quasi left $m$-invertible
operators, $n$-quasi $m$-isometric operators, $n$-quasi $m$-selfadjoint
operators and $n$-quasi $(m,C)$ symmetric operators (for some conjugation $C$
of $\H$). It is proved that $S^n$ is the perturbation by a nilpotent of the
direct sum of an operator $S_1^n=(S|_{\overline{S^n(\H)}})^n$ satisfying
$d^m_{T_1,S_1}(I_1)=0$, $T_1=T|_{\overline{S^n(\H)}}$, with the $0$ operator;
if also $S$ is left invertible, then $S^n$ is similar to an operator $B$ such
that $d^m_{B^*,B}(I)=0$. For power bounded $S$ and $T$ such that $ST^*-T^*S=0$
and $\triangle_{T,S}(S^{*n}S^n)=0$, $S$ is polaroid (i.e., isolated points of
the spectrum are poles). The product property, and the perturbation by a
commuting nilpotent property, of operators $T, S$ satisfying $d^m_{T,S}(I)=0$,
given certain commutativity properties, transfers to operators satisfying
$S^{*n}d^m_{T,S}(I)S^n=0$.
