Solution of a tridiagonal operator equation

Let H be it separable Hilbert space with an orthonormal basis {e(n)/n is an element of 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 {parallel to T(n)(-1)e(n)parallel to} and {parallel to T-n*(-1) e(n)parallel to} 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 {parallel to T-n(-1) e(n)parallel to} and {parallel to T-n*(-1)e(n)parallel to} in terms of the entries of the matrix of T. (c) 2005 Elsevier Inc. All rights reserved.