Prove that there $B : H\to H $ bounded such $ B^n = A $.

Question

Let $ A : H\to H $ a compact self-adjoint operator. Suppose $ A $ is positive. let $ n \geq 2 $. Prove that there is $B : H\to H $ bounded such $ B^n = A $.

Answer 1

By the Spectral Theorem, $A$ is of the form $$ A=\sum_j\lambda_j\,P_j, $$ where $\lambda_j\in\mathbb R$ and $P_j$ is a finite-rank projection for all $j$.

Now choose numbers $\mu_j\in\mathbb C$ with $\mu_j^n=\lambda_j$ and define $$ B=\sum_j\mu_jP_j. $$