Journal of Operator Theory

Volume 61, Issue 1, Winter 2009 pp. 119-131.

An extended class of integrable operators

Authors: Lev Sakhnovich
Author institution: 735 Crawford ave. New York, 11223, USA

Summary: We consider an extended class of the integrable operators. This class is connected with the class of the Riemann--Hilbert problems. We show that the solution $W(z)$ of the Riemann--Hilbert problem coincides with the monodromy matrix of the corresponding differential system. It follows from this result that $W(z)$ can be represented in two forms: 1. In the form of the transfer matrix function. 2. In the form of the multiplicative integral. The analogues of the Plemelj formulas are deduced in the paper for limiting values of the multiplicative integral $W(z)$.

Keywords: Riemann--Hilbert problem, differential system, monodromy matrix, multiplicative integral, triangular factorization.

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