Let $H$ be an Hilbert space, and let $T$ be a self-adjoint, positive (and therefore bounded) operator $H \to H$, with $||T||<2012$. Let $P$ be a polynomial with real coefficients such that the minimum of $P$ on $[-2012,2012]$ is positive. Consider the operator $S=P(T)$. It is also self-adjoint and bounded. I ask : must it necessarily be positive (i.e. do we have $(Sx,x) \geq 0$ for every $x$)?
If $T$ is diagonalizable by an orthogonal basis $\cal B$, then $S$ will also be diagonal relatively to $\cal B$, and the eigenvalues of $S$ will be the numbers $P(\lambda)$, where $\lambda$ is an eigenvalue of $T$. So the answer is yes in this case.
This covers the case when $H$ is finite-dimensional. When $H$ is infinite-dimensional, however, things are not so clear (at least to me).