The Research About Invariant Subspace Of Operator Theory

The study on invariant subspace problem has always been a hot issue in operator theo-ry.Scholars have spent many years studying and thinking and have achieved a lot of excellent results.However,there is still a long way to go to solve the difficult problem.Invariant sub-space were defined by von Neumann in 1935 for natural problems of spectral theory.It soon became clear that this object can replace the fundamental concept of finite-dimensional lin-ear analysis.For a long time,however,the geometric description of the invariant subspace did not produce a signature conclusion.In 1954,Aronszajn and Smith proved a proposition by the approximation of the finite rank operator.In Banach space,arbitrary bounded and compact operators have nontrivial invariant subspaces.After 19 years,Lomonosov had used Schauder fixed-point theorem to prove anther proposition.If a non-zero compact operator in Banach space can be exchanged with operator T,then T has non-trivial invariant subspace.On the other hand,people have proved that the concept of invariant subspace is directly related to some established disciplines.Chapter one of this paper we will introduce respec-tively properties about invariant subspace of completely continuous operators,shift operators and rational Toeplitz operator.In the chapter two we will know two special spaces,spaces of analytic functions and Bergman space Banach.There are some results about their invariant subspaces.