Operator norm of the self adjoint part

Question

Let $A=\frac{1}{2}(T+T^\ast)$ be the self adjoint part of a bounded operator $T$ on a Hilbert space. Let $f$ be a polynomial. When do we have $\|f(A)\|\leq \|f(T)\|$ in the operator norm?

The inequality $\|A\|\leq \|T \|$ follows from the triangle inequality and $\|T\|=\|T^\ast \|$, but I can't convince myself of even $\|A^2\|\overset{?}{\leq}\|T^2\|$.

Perhaps at least $\|f(A)\|\leq c_{T,f}\|f(T)\|$ for a positive constant if $f$ is suitable?

Answer 1

Consider $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by the matrix $\left( {\begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} } \right)$, which has self-adjoint part $A = \frac{1}{2}\left( {\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)$. Then $T^2 = 0$, but $A^2 = \frac{1}{4} I$, so the inequality $\lVert A^2 \rVert \leq \lVert T^2 \rVert$ is wrong in general.