The operators such as aP PaI − , and integral operators with weak singularities are studied in the work. It is proven that the operators and are totally continuous (or compact) in spaces with weights in one and only one case, when the function is continuous on the contour of integration. As a corollary, it is shown that the factor-algebra generated by singular operators with piecewise continuous coefficients is not comutative and the
symbol on that algebra is a matrix-function.
