Das, Soma2023-09-132023-10-202023-09-132023-10-202023ROLL NO.186123016https://gyan.iitg.ac.in/handle/123456789/2458Supervisor: Chattopadhyay, ArupIn this dissertation, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector valued Hardy space. Using this characterization we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space. Moreover, as an application, we also identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases using our characterization in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector valued Hardy space.enShift OperatorNearly Invariant SubspacesToeplitz OperatorBlaschke ProductHankel OperatorSchmidt SubspacesThesis