Solution of a tridiagonal operator equation

Abstract

Let H be a separable Hilbert space with an orthonormal basis {e_n/n ∈ N}, T be a bounded tridiagonal operator on H and T_n be its truncation on span ({e₁, e₂,..., e_n}). We study the operator equation Tx = y through its finite dimensional truncations T_nxⁿ = y_n. It is shown that if {||T⁻¹_ne_n||}and{||T^∗−1_ne_n||} are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence {T⁻¹_n}. We also give sufficient conditions for the boundedness of {||T⁻¹_ne_n||} and {||T^∗−1_ne_n||} in terms of the entries of the matrix of T.