  In this paper we prove H\"ormander-Mihlin multiplier theorems for
pseudo-multipliers associated to the harmonic oscillator (also called the
Hermite operator). Our approach can be extended to also obtain the
$L^p$-boundedness results for multilinear pseudo-multipliers. By using the
Littlewood-Paley theorem associated to the harmonic oscillator we also give
$L^p$-boundedness and $L^p$-compactness properties for multipliers.
$(L^p,L^q)$-estimates for spectral pseudo-multipliers also are investigated.
