Function of linear operator

Question

Under what conditions on a given function $f$ and a self ajoint operator $A$ we have $f(A)$ is self ajoint?

How about the norm of $f(A)$? Is it possible to exprime it in term of the norm of $A$?

Answer 1

Consider $A\in \mathcal{L}^{SYM}(H)$,(so $A$ is symmetric and bounded) $H$ a $\mathbb{C}$-Hilbert space. Let $\sigma(A)\subset \mathbb{C}$ be the spectrum of $A$ i.e. $\sigma(A) = \{\lambda \in \mathbb{C}\ s.t. \ \lambda I - A \not\in GL(H)\}$. In this case*, a Borelian Calculus can be developed, it means that given $f\in L^\infty(\sigma(A), \mathbb{C})$ borelian you can construct $f(A)\in \mathcal{L}(H)$ in a reasonable way. The simplest case being when $f$ is a polynomial in $\mathbb{C}[z]$. In this case $f(z) = \sum_{i=0}^n a_nz^n$ gives $f(A)= \sum_{i=0}^n a_nA^n$. The general construction is a generalization of this one firstly to $C^0(\sigma(A), \mathbb{C})$ (exploiting the density of polynomials wrt the supremum norm) then to Borelian functions (not so straightforward).

Notice that already with polynomials if $p\in \mathbb{C}[z]$ is not real valued, i.e. $\overline{p}(z)\neq p(z) \ \forall z\in \sigma(A)$ then $p(A)$ is not symmetric, for example $i A$ is bounded but not symmetric $(iA)^* = -i A^* = -iA\neq iA$ (prove it exploiting the sesquilinearity of $\langle\cdot,\cdot\rangle$). It holds in general that $f(A)^*= \overline{f}(A)$ so $f(A) $ is symmetric only if $f$ is real.

For what regards the norms, for polynomials $||p(A) ||_{\mathcal{L}(H)} = ||p||_{\infty,\sigma(A)}$ (by the spectral mapping theorem) and since the continuous calculus is done extending by density this isometry to $C^0(\sigma(A),\mathbb{C})$ it holds also for continuous functions i.e. $||f(A) ||_{\mathcal{L}(H)} = ||f||_{\infty,\sigma(A)}$. Unfortunately for the Borelian extension I have only seen the proof that $||f(A) ||_{\mathcal{L}(H)} \leq ||f||_{\infty,\sigma(A)}$, so I don't know if there is a general formula for the norm of $f(A)$ when $f\in L^\infty$ borelian.

A book where you can find this theory developed is for example the Kirillov-Gvishiani "Theorem and Problems in Functional Analysis". It is a very nice book because it contains problems,hints to solve them and if you are interested in Quantum Mechanics there is also an introductory section on it.

*Actually it can be developed also in the case when $A$ is only densely defined and self-adjoint, you don't need it to be bounded.