# Self-adjoint operator proportional to identity if and only if the support of the spectral measure is a singleton

Let A be a self-adjoint operator on a Hilbert space $\mathcal{H}$, and let $P_A$ be the projection valued measure (spectral measure) obtained from the spectral theorem, such that

$A=\int \lambda\ dP_A(\lambda)$

How do I show that

$A=\lambda I$ $\ \ \$ $\Leftrightarrow$ $\ \ \$ $\text{supp}P_A=\{\lambda\}$

• Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. – José Carlos Santos Mar 7 '18 at 9:53
• Of course i did not think of that! Well I think I know one of the ways. If $A=\lambda I$ then the spectrum is $\{\lambda\}$ and the spectral measure is defined on this spectrum, hence supp$P_A=\{\lambda\}$. But for the other direction i do not know what to do? – Simon Mar 7 '18 at 10:02