dc.description.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. | en_US |