In my functional analysis lecture we introduced the continuous functional calculus on $\sigma(T)$ if $T$ is a self-adjoint operator. Then the Riesz representation theorem gives us that $l_x(f):=\langle f(T)x,x \rangle = \int_{\sigma(T)} f d\mu_{x}.$ Now, I was wondering if this measure $\mu_{x}$ is always positive or if there is a condition that tells us when this measure is positive?
I suspect that it is always a positive measure, cause from the spectral theorem we have $$\langle f(T)x,x\rangle = \int_{\sigma(T)} f \underbrace{\langle dE(x),x\rangle}_{d\mu_{x}}.$$
Yes, it is positive.
Let $f : \sigma(T) \to [0,\infty)$ be any nonnegative continuous function. I claim that $f(T)$ is a nonnegative operator. Then it follows that $\int f \,d\mu_x = \langle f(T) x , x \rangle \ge 0$. As this holds for all nonnegative continuous $f$, it must be that $\mu_x$ is a positive measure.
To prove the claim, let $g(t) = \sqrt{f(t)}$ which is also a continuous function. Then $f = g \bar{g}$, so since the functional calculus is a *-algebra homomorphism we have $f(T) = \bar{g}(T) {g}(T) = g(T)^* g(T)$. Thus $$\langle f(T) x, x \rangle = \langle g(T)^* g(T) x, x \rangle = \langle g(T) x, g(T) x \rangle = \|g(T) x\|^2 \ge 0.$$