Let $f : [0,\infty) \longrightarrow \mathbb R$ be a continuous function. Let $\mathcal H$ be any Hilbert space and let $\mathbb H$ and $\mathbb P$ be the spaces of self-adjoint operators on $\mathcal H$ and positive operators on $\mathcal H,$ respectively. We denote by $\widetilde {f},$ the induced map from $\mathbb P$ to $\mathbb H.$
Suppose $f$ is convex on $[0, \infty),$ and suppose $\mathcal H$ is a finite dimensional vector space. Suppose $A \in \mathbb P$ and $u$ is a unit vector in $\mathcal H.$ Show that
$$f \left (\left \langle u, Au \right \rangle \right) \leq \left \langle u, f(A) u \right \rangle.$$
Does the above remain true if $\mathcal H$ is an infinite dimensional space? Justify your answer.
This question was appeared in the final semestral examination in functional analysis which I am unable to do properly. I have proved the above inequality for finite-dimensions but for infinite-dimensions I don't have any idea. Could anyone please help me in this regard?
Thanks for investing your valuable time in reading my question.
The functional calculus for self-adjoint operators is known to be monotone: If $f,g:\sigma(A) \to {\mathbb R}$ are continuous and $g \le f$ on $\sigma(A)$, then $g(A) \le f(A)$. If $f:[0, \infty) \to {\mathbb R}$ is continuous and convex and $u \in {\cal H}$ is a unit vector, then there is a supporting affine function in $\langle u,Au \rangle$, that is a function $g:{\mathbb R} \to {\mathbb R}$, $$ g(t)=f(\langle u,Au \rangle)+ \alpha(t-\langle u,Au \rangle) $$ with $g(t) \le f(t)$ on $[0, \infty)$. Now $g(A) \le f(A)$, thus $$ f(\langle u,Au \rangle)=f(\langle u,Au \rangle) \langle u,u \rangle + \alpha (\langle u,Au \rangle -\langle u,Au \rangle \langle u,u \rangle)= \langle u,g(A)u \rangle \le \langle u,f(A)u \rangle. $$