Invariant Subspaces In The Spaces Of Analytic Functions

In the first part of this paper, we state invariant subspaces of Toeplitz operators in the spaces of analytic functions such as Hardy spaces, Dirichlet spaces and Bergman spaces. We see that z - invariant subspaces of these spaces are in one-to-one correspondence with the wandering subspaces of Mz, where the correspondence is given by M = [M zM]. We see that the nonzero wandering subspace M zM of Hardy space H2 or Dirichlet space D1, is one-dimensional, while the dimension of the nonzero wandering subspace M zM of Bergman space L2a may is any positive integer or And we see that a g - invariant subspace N in Hardy space H2 is also one-to-one correspondence with the wandering subspace N gN of Tg, where the correspondence is given by N =[N gN]g. In the second part of this paper, we discuss the reducing subspaces of an isometry operator on a Hilbert space, and obtain a construction of the reducing subspace of the Toeplitz operator TB on H2, where B is a Blaschke product.