Invariant and Hyper-invariant Subspaces of Some Classes of Operators in Hilbert Spaces

Abstract

Direct sum decomposition of bounded linear operators gives a way of studying complicated operators since it decomposes these operators into parts whose structures are user friendly. It is known that some operators are not decomposable, however, every reducible operator is decomposable. Invariant subspaces of an operator, their classi cation play an explicitly central role in studying complicated operators, for it is known that, every operator with a nontrivial invariant subspace is reducible. Reducing subspaces are special invariant subspaces which are useful in the direct sum decomposition. The motivations behind the study of invariant subspaces come from the interest in the structure of an operator and from approximation theory to a wide variety of problems in physics(quantum theory), computer science (data mining) and chemistry(lattice theory of crystal analysis). In this thesis, we have investigated the existence of invariant and hyperinvariant subspace for some operators such as quasinormal,nilpotent, quasinilpotent, hyponormal, paranormal among others. We have shown that quasinormal as well as nilpotent operators have nontrival invariant subspace. Quasinilpotent operators have nontrivial invariant subspaces if the operator and its adjoint satisfy Single Value Extension Property. Conditions in which hyponormal and higher classes have nontrivial invariant subspaces are given. We have studied the structure of lattices of some of the operators in relation to certain equivalence classes. We have shown that self adjoint operator T and its adjoint have equal lattices. It has also been shown a normal operator and its adjoint have isomorphic invariant subspace lattices but fails for quasinormal operators. It has been shown that isomorphism of hyperlattices of operators does not imply quasisimilarity nor similarity of operators.