# Square Root of Positive Operator

1. Apr 12, 2013

### Shoelace Thm.

1. The problem statement, all variables and given/known data
Suppose $U = T^2 + \alpha T + \beta I$ is a positive operator on a real inner product space V with $\alpha^2 < 4 \beta$ . Find the square root operator S of U.

2. Relevant equations

3. The attempt at a solution
Isn't this just the operator $S \in L(V)$ such that $S e_k = \sqrt{ \lambda_k } e_k$, where the $e_k$ form an orthonormal basis of eigenvectors of U? Can we get anymore specific here than the definition?