Square Root of Positive Operator

[Shoelace Thm.]

Homework Statement

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.

Homework Equations

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?

 

[mighty2000]

Yes, you are correct. The square root operator S can be defined as the operator in L(V) such that S e_k = √(λ_k) e_k, where the e_k form an orthonormal basis of eigenvectors of U. However, it is important to note that this definition only works if U has real eigenvalues. If U has complex eigenvalues, then the definition of S becomes more complicated and is given by S = √(U) = U^(1/2) = ∑_k √(λ_k) P_k, where P_k is the projection onto the eigenspace associated with eigenvalue λ_k. So, in this case, the square root operator is a combination of the projection operators onto the eigenspaces of U.