| Abstract: | An operator integral, referred to as the amplitude integral (AI) and used in the BC-method (based on boundarycontrol theory) for solving inverse problems, is systematically studied. For a continuous operator and two families of increasing subspaces, the continual analog of the matrix diagonal in the form of an AI is introduced. The convergence of the AI is discussed. An example of an operator with no diagonal is provided. The role of the diagonal in the problem of triangular factorization is elucidated. The well-known result of matrix theory stating the uniqueness of triangular factorization with a prescribed diagonal is extended. It is shown that the corresponding factor can be represented in the AI form. The correspondence between the AI and the classical representation of the triangular factor of an operator that is a sum of the identity and a compact operator is established.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 45–60.This work was supported by INTAS under grant INTAS 93-1815 and by the Russian Foundation for Basic Research under grant 96-01-00666. The authors are grateful to Prof. S. N. Naboko for fruitful discussions and consultations. |
