ISSN 0001-4346, Mathematical Notes, 2018, Vol. 104, No. 3, pp. 404–416. © Pleiades Publishing, Ltd., 2018. Original Russian Text © A. G. Kamalyan, I. M. Spitkovsky, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 3, pp. 407–421. On the Fredholm Property of a Class of Convolution-Type Operators A. G. Kamalyan 1,2* and I. M. Spitkovsky 3** 1 Yerevan State University, Yerevan, 375025 Armenia 2 Institute of Mathematics, National Academy of Sciences, Yerevan,375019 Armenia 3 New York University Abu Dhabi, Abu Dhabi, United Arab Emirates Received December 5, 2017; in final form, February 3, 2018 Abstract—The notions of the L -convolution operator and the L -Wiener–Hopf operator are introduced by replacing the Fourier transform in the definition of the convolution operator by a spectral transformation of the self-adjoint Sturm–Liouville operator on the axis L . In the case of the zero potential, the introduced operators coincide with the convolution operator and the Wiener–Hopf integral operator, respectively. A connection between the L -Wiener–Hopf operator and singular integral operators is revealed. In the case of a piecewise continuous symbol, a criterion for the Fredholm property and a formula for the index of the L -Wiener–Hopf operator in terms of the symbol and the elements of the scattering matrix of the operator L are obtained. DOI: 10.1134/S0001434618090080 Keywords: the operator L -Wiener–Hopf, singular integral operator, Fredholm property. Dedicated to the memory of Nikolai Karapetovich Karapetyants, a talented mathematician and a remarkable man. 1. PROBLEM STATEMENT Let L be the maximal symmetric operator generated by the differential expression (ℓy)(x)= −y ′′ + p(x)y(x) with real potential p satisfying the inequality ˆ ∞ −∞ (1 + |x|) |p(x)| dx < ∞. The operator L is self-adjoint, and its discrete spectrum consists of a finite number ν of simple eigenvalues (see [1]–[3]). In what follows, given a linear space X, we use X n (respectively, X n×n ) to denote the set of n-columns (respectively, of n × n matrices) with elements in X. We denote the identity operator by I , every time specifying the space on which it acts. Given an operator A acting from a linear space X to a linear space Y , we denote the operator diag(A,...,A): X n → Y n by the same symbol A; i.e., the action of A on X n is assumed to be entry-wise. Let m(a) denote the operator acting on function spaces as multiplication by a function (matrix function) a: (m(a)y)(x)= a(x)y(x). * E-mail: armen.kamalyan@ysu.am, kamalyan_armen@yahoo.com ** E-mail: ims2@nyu.edu, imspitkovsky@gmail.com 404