Operator identities corresponding to inverse problems for Dirac systems

Abstract

The structured operators and corresponding operator identities, which appear in inverse problems for the self-adjoint and skew-selfadjoint Dirac systems with rectangular potentials, are studied in detail. In particular, it is shown that operators with the close to displacement kernels are included in this class. A special case of positive and factorizable operators is dealt with separately. MSC(2010): 45H05, 47G10, 34L40, 47A68.

Recovery of the Dirac system from the rectangular Weyl matrix function

“…For that we recall first that S > 0 and Φ 1 is boundedly differentiable in Lemma 2.4. Therefore, using Theorem 2.5 from [15] we get the statement below.…”

“…Proposition 5.3[15] Let Φ 1 (x) be an m 2 × m 1 matrix function, which is boundedly differentiable on the interval [0, l]. Then the operator S, which is given by(3.8), satisfies the operator identity (2.18), where Π :…”

Inversion of the convolution operators on a rectangular

“…The method of operator identities, which was introduced in [17] (see also [16]), may be successfully used for the inversion of various other structured operators. See, for instance, [4,14] and a more detailed discussion with references in [15,Appendix D].…”

Inversion of the Toeplitz-block Toeplitz matrices and the structure of the corresponding inverse matrices

“…The method of operator identities, which was introduced in [30,31], may be successfully used for the inversion of various other structured matrices and operators. See, for instance, [10] and [29,Appendix D]. Note also that relations of the form S * T S = T were used for the study of Toeplitz operators in the seminal work [8].…”

“…The inversion of Toeplitz and Toeplitz-block Toeplitz (TBT) matrices and their continuous analogs is actively studied in the recent years as well (see e.g. [1,2,4,5,9,10,17,24,33,38] and references therein). However, in spite of some interesting recent and older works [12, 16-19, 22, 23, 37] on the inversion of TBT-matrices and of convolution operators in multidimensional spaces, the structure of the corresponding inverse matrices and operators (and the way to recover these inverses from some minimal information) remained unknown.…”