Steepest Descent for Singular Linear Operator Equations

Let H be a Hilbert space, T a bounded linear operator on H into H such that the range of T is closed. Let $T^ * $ denote the adjoint of T. In this paper, we establish the convergence of the method of steepest descent to a solution of the equation $T^ * Tx = T^ * b,\, b \in H$, for any initial approximation $x_0 \in H$. We also show that the method converges to the unique solution with minimal norm if and only if $x_0 $ is in the range of $T^ * $.