Geodesic
On a triangular factorization of positive operators
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 26, Tome 239 (1997), pp. 45-60

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The paper deals with an operator construction (so-called Amplitude Integral) working in the BC-method for the inverse problems. A continual analog of a matrix diagonal is introduced for a continuous operator and a pair of extending families of subspaces. The analog is represented via the AI. Its convergence is descussed; we demonstrate an example of the operator which doesn't possess a diagonal. A role of a diagonal in the problem of triangular factorization is clarified. A well-known result of the theory of matrices is that a factorization with fixed diagonal is unique. In the paper this result is generalized on a class of positive operators, the corresponding factor being given in the form of the AI. Relations between the AI and the classical operator integral (M. Krein and oth.) used to factorize operators $\mathbf1+{}$K (with compact K) are established. 
@article{ZNSL_1997_239_a3,
     author = {M. I. Belishev and A. B. Pushnitskii},
     title = {On a triangular factorization of positive operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {45--60},
     publisher = {mathdoc},
     volume = {239},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_239_a3/}
}
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M. I. Belishev; A. B. Pushnitskii. On a triangular factorization of positive operators. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 26, Tome 239 (1997), pp. 45-60. http://geodesic.mathdoc.fr/item/ZNSL_1997_239_a3/