Galerkin's method for operator equations with nonnegative index — With application to Cauchy singular integral equations

We consider solving operator equation (1) Hu + Ku = f, where H and K are bounded linear operators between two real Hilbert spaces H₁ and H₂. Operator H is assumed to have a finite-dimensional nullspace N(H) and a bounded right inverse H¹:H₂ → H₁ and K is compact. It follows that dim(N(H + K)) = dim(N(H)), so that to obtain uniqueness the m additional conditions (2) 〈u,φ_k〉₁ = b_k, k=1, 2, h.,dim(N(H)) = m are imposed, where the {φ_k}_{k = 1}^m are an orthonormal basis for N(H). To solve (1) and (2), these equations are converted to an equivalent equation of the second kind to which Galerkin's method is applied using the basis $\text{{φ}{}_1\text{, φ}{}_2\text{, …, φ}{}_{\mathrm{m}}\text{, ¦φ}{}_{\mathrm{m\; +\; 1}}\text{,…, φ}{}_{\mathrm{n}}\text{,…}}$. It was shown that this method is equivalent to the method of weighted residuals when $\text{H}{}^1\text{= H}{}^1$ (the adjoint of H). The results are applied to obtain convergence proofs of some numerical methods for solving several classes of Cauchy singular integral equations whose kernels are only square integrable.