Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line 0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .