On the properties of a class of integral operators in the space Lp

In the spaceL _p (?),p > 1, we consider the operatorA=a? +bS? +cP? +T?, wherea(t), b(t), and c(t) are piecewise-continuous functions on the contour ?, T is a completely continuous operator, P?=1/2πi∫ ?(τ) dτ/? τ ?t ? 1, S?=1/gpi∫ ?(τ) dτ/? τ ? i, ? is a closed convex Lyapunov contour having no rectilinear portions. We study the properties of the operator P and we show that the Noether property conditions and the index of the operator A do not depend on the term cP.