Ill-conditioned analytic boundary value problems are reduced to one integral equation, which is called commutative: \[ [S,a] \varphi=f \quad\text{on} \quad\Pi,\tag{1} \] where \([S,a]= Sa-aS\) is the commutator of the operators of singular integration on the unit circle \(\Pi\) and of multiplication by the function \(a(t)\). An approximate solution to the boundary value problems is constructed by applying a modified Tikhonov regularization procedure to solve the integral equation (1). In implementing the procedure, two representations are obtained for the regularized solution to the commutative equation and therefore, to the initial problems. One of these is written in terms of the factorization of two Hermitian positive definite matrices, and the second expresses the approximate solution through the resolvents of selfadjoint integral operators.
The two representations of the approximate solution to equation (1) make it possible to express the deep interrelation between selfadjoint integral operators and Hermitian matrices in an explicit form. The formulas for resolvents are also obtained in terms of the factorizations of the corresponding matrices.