On the triangular factorization of isomorphisms

The paper deals with an operator construction (so-called the Amplitude Integral) working in the BC-method for dynamical inverse problems. The AI is applied to the problem of the triangular factorization, the class of factorized operators being isomorphisms of the Hilbert space. A continual analog of matrix diagonal is introduced. Uniqueness of the factorization in which one of the factors has the prescribed diagonal is established. Under additional assumptions on operator, the representation of the factors through the AI is obtained. This representation gives efficient tool of the factorization. Some of the obtained results generalize the classical ones concerning to the factorization of operators of the class “unit plus compact.”