  We report on a set of studies concerning the description of New Physics (NP)
effects characterized by a scale much higher than the electroweak scale. We
concentrate on the residual effects described by an effective lagrangian
involving only bosonic fields and restrict to the case that the Higgs field is
described linearly, so that the Higgs particle exists. It has already been
emphasized that this lagrangian may be represented by eight $dim=6$ operators;
namely the blind CP-conserving operators $\O_W$, $\O_{B\Phi}$, $\O_{W\Phi}$,
$\O_{UB}$, $\O_{UW}$, the ''superblind'' ${\cal O}_{\Phi 2}$ and the two
CP-violating ones $\ol{\O}_{UB}$ and $\ol{\O}_{UW}$. To each operator and for
any given value of its coupling, we associate an unambiguous NP scale
determined by the energy where unitarity is saturated. Our study of the
possible NP tests realizable at a future linear $e^+e^-$ collider concentrates
on $(e^+e^-\to HZ ~,~ H\gamma)$, and on $\gamma\gamma$ collisions producing
boson pairs ($\gamma\gamma \to W^+W^-$, $ZZ$, $Z\gamma$, $\gamma\gamma$, $HH$),
or a single Higgs ($\gamma\gamma \to H$). The sensitivities to the various
operators involved in the effective lagrangian are estimated. We find that the
testable NP scales vary from a few tens of TeV in the boson pair processes, up
to 65 TeV in single H production. Ways to disentangle the various operators are
also proposed. In this respect a comparison of the NP effects in the various
boson pair processes, like \eg\@ a comparison of $e^+e^-\to HZ$ versus
$e^+e^-\to H\gamma$, should be very fruitful. This is applied also to the study
of the Higgs branching ratios, and especially the ratio
$\Gamma(H\to\gamma\gamma)/\Gamma (H\to\gamma Z)$.
