Let $H$ and $K$ be Hilbert spaces and $T:H \to K$ be a compact operator.
Does there exists nets $e_{\lambda}$ and $f_{\lambda}$ of finite rank operators such that the net $e_{\lambda}f_{\lambda}^*T \in K(H,K)$ converges to $T$ in norm Topology? Here $f_{\lambda}$ denotes adjoint of $f_{\lambda}$ in $K(H,K)$
Since finite rank operators forms approximate identity for compact operators therefore it seems it should be true. Any ideas for proof or counterexample?
Yes and in fact you can choose it to be a sequence.
Since $T$ is compact, $TT^*$ and $T^*T$ are compact self-adjoint, so you get orthogonal eigenspace decomposition with the same eigenvalues (ignoring the kernel which we don't really care here). Let $e_n=f_n\colon H\to K$ be identifying eigenspaces with eigenvalue $>\frac1n$ via $\frac1{\sqrt{\lambda}}T$ and $0$ on the other eigenspaces, so $e_nf_n^*$ is orthogonal projection onto eigenspaces. Now it is routine to check $e_nf_n^*T\to T$ in norm.