Suppose $T$ is a positive invertible element in $B(H)$. Does there exits a sequence $T_n$ of positive invertible operators with finite spectrum such that $T_n$ converges to $T$?
I know the following fact:Let $H$ be a complex Hilbert-space and $A:H\to H$ be a normal operator. There is a unique spectral measure $\Phi$ on the spectrum $\sigma(A)$ such that $f(A) = \int_{\sigma(A)} f d \Phi$ for any continuous function $f$ on the spectrum $\sigma(A)$.
Then $T=\int_{\sigma(A)} \iota d \Phi$,where $\iota (z)=z$.How to choose the sequence $T_n$ such that $\sigma(T_n)$ is finite and $T_n \to T$?