Let H 1 and H 2 be Hilbert spaces, let T ∈ B ( H...
Let and be Hilbert spaces, let . Suppose that is finite-dimensional and that is closed in . Prove that is finite-dimensional for each .
1. Define Hilbert space as direct sum of two complemented subspaces
2. For compact operator use Hilbert Schmidt decomposition
Main idea is to prove that intersection of Spectrum of such $T and compact K is finite