Common invariant subspaces of pairs of operators on a hilbert space.

Let H be an infinite dimensional separable complex Hilbert space. Let , A єB(H) is the Banach algebra of all bounded linear operators on H. A closed linear manifold (subspace) M of H is said to be a non-trivial invariant subspace under A if 0≠M≠H and AMcM. On the other hand, a common non-trivial invariant subspace (CNIS) of A, BєB(H) is a non-trivial subspace which is invariant under A and B. It is known that not each pair of operators in B(H) have CNIS Therefore, in this work, we search for conditions on a pair of operators under which they have a CNIS. We begin by studying the existence of a common invariant subspace of an operator TєB(H) with its adjoint. Second, we shed light on the existence of a CNIS for operators with finite rank commutator. In particular, we look for additional assumptions on a pair of commutating operators to get a CNIS of such operators. Also, for rank one commutator, we give Laffeye's theorem with some new consequences and some"" extensions that include pairs of operators with finite rank commutators. Finally, we investigate CNIS for algebraic operators, in particular, operators that satisfy polynomial equations of degree 2. Among our results in this connection is the following: Let A, BєB(H) such that rank [A,B]