On Two Families of Operator Equations, Vector Continued Fractions \(K\), and Related Band Operators

Abstract

We study two families of equations in the algebra of bounded operators in a Hilbert space and show how the theory of block band operators and vector continued fractions with operator coefficients can be applied for obtaining their solutions. In particular, for each of such equations, we find a corresponding block band operator and show that, under an additional condition related to its spectrum, an exact solution of this equation can be expressed through the elements of the Weyl matrix of this operator. Also, using the reduced versions of a block band operator of this kind, we explicitly construct approximate solutions of the equation and show that they converge to the exact solution at a geometric rate.