The convenience of a boson basis used in conjunction with a boson mapp
ed hamiltonian may sometimes be complicated by the appearance of spuri
ous states in the spectrum. Under ideal circumstances where either no
truncation of the boson Fock space is introduced, or where truncation
involves a set of completely decoupled collective degrees of freedom o
nly, the identification and role of such spurious states are well unde
rstood and analysed. However, for situations where truncation involves
degrees of freedom which are not completely decoupled, no analogous s
ystematic analysis has yet been given. A formalism which suggests itse
lf for such an analysis, is effective operator theory. The energy-inde
pendent effective interaction approach of Suzuki and Lee seems to be w
ell suited for this purpose. We first extend this approach beyond inte
ractions only, and construct the effective operator corresponding to a
n arbitrary operator. We then show how the formalism is applied in the
present context of identifying spurious states. In particular, we app
ly the extended procedure to the generalized SO(5) model, to a single-
j proton-neutron model, and also to the multi-j similarity-transformed
Dyson mapping. We also compare the present approach to an alternative
energy-independent effective-operator approach which is based on the
use of a Majorana-like operator in conjunction with the SO(2n) Dyson b
oson mapping. The present application is also linked to applications i
n the derivation of effective shell-model interactions.