Powers of compact operators

Question

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I see that powers of bounded operators are bounded. But I am unsure how to prove the corresponding statement for compact operators.

Answer 1

Using the spectral theorem: A selfadjoint operator $T \ne 0$ on a Hilbert space is compact iff $$ T = \lambda_1 E_{1} + \lambda_2 E_{2} + \cdots, $$ where $\{ E_{j} \}$ is a finite or countably infinite set of disjoint orthogonal projections onto finite-dimensional subspaces, and the sequence $\{ \lambda_{j} \}$, if infinite, converges to $0$. (The convergence of the sum, if infinite, is in the strong operator topology.)

With that spectral characterization, you can see that powers of $T$ are compact: $$ T^{\alpha} =\lambda_1^{\alpha}E_{1}+\lambda_2^{\alpha}E_{2}+\cdots. $$