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 ({ e 1, e 2, … , e n }). We study the operator equation Tx = y through its finite dimensional truncations T n x n = y n . It is shown that if { ‖ T n - 1 e n ‖ } and { ‖ T n ∗ - 1 e 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 n - 1 } . We also give sufficient conditions for the boundedness of { ‖ T n - 1 e n ‖ } and { ‖ T n ∗ - 1 e n ‖ } in terms of the entries of the matrix of T.