  We identify the norm of the semigroup generated by the non-self-adjoint
harmonic oscillator acting on $L^2(\Bbb{R})$, for all complex times where it is
bounded. We relate this problem to embeddings between Gaussian-weighted spaces
of holomorphic functions, and we show that the same technique applies, in any
dimension, to the semigroup $e^{-tQ}$ generated by an elliptic quadratic
operator acting on $L^2(\Bbb{R}^n)$. The method used --- identifying the
exponents of sharp products of Mehler formulas --- is elementary and is
inspired by more general works of L. H\"ormander, A. Melin, and J. Sj\"ostrand.
