Problem 8.11 in Young's book (An introduction to Hilbert space)

Question

I've tried to prove this problem which appears in Young's Book:

Let $K$ be a compact Hermitian operator on a Hilbert space $H$ and let the kernel of $K$ be $\{0\}$. Show that there is a sequence $(K_n)$ of bounded linear operators on $H$ such that $K_nKx \rightarrow x$ and $KK_nx \rightarrow x$ as $n \rightarrow \infty.$

I think $\displaystyle K_n = \sum_{i=0}^n (I-K)^i$ but I can't prove it is a required sequence. Could anyone please help me?

Answer 1

By the spectral theorem, $H$ has a countable orthonormal basis consisting of eigenvectors $e_i$ of $K$, with corresponding eigenvalues $\lambda_i>0$. Define $K_n$ by $K_n e_i=\lambda_i^{-1}e_i$ for $1\leq i\leq n$, and $K_n e_i=0$ for $i>n$. Given an element $x=\sum_{i=1}^\infty \alpha_ie_i$, we have $KK_nx=K_nKx=\sum_{i=1}^n \alpha_ie_i \to x$, as desired.