Inveriant Subspaces and Some Structure Theorems

First we shall give some historical background about the problem of invariant subspaces for bounded operators on Hilbert and Banach spaces. As is mentioned in the fundamental paper of Aronszajn and Smith (1954), the first result about the existence of invariant subspaces was proved by J. von Neumann: If T is a bounded linear and compact operator on a Hilbert space H, then there exists a closed subspace H₁ ⊂ H with the property that if x ∈ H₁, then Tx ∈ H₁. In the above assertion we suppose that H₁ has the property