Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities.pdf

Compl. anal. oper. theory 2 (2008), 549–567 c? 2008 Birkha?user Verlag Basel/Switzerland 1661-8254/040549-19, published online July 15, 2008 DOI 10.1007/s11785-008-0066-x Complex Analysis and Operator Theory Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities Aleksandr Karelin Abstract. In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix char- acteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t?1, were found. Conditions for the invertibility of the matrix characteristic singular in- tegral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Mathematics Subject Classification (2000). Primary 47G10, 45E99, 54F15. Keywords. Singular integral operator, matrix characteristic operator, piece- wise constant coefficients, invertibility. 1. Introduction We denote by [B1, B2] the set of all bounded linear operators mapping the Banach space B1 into the Banach space B2, [B1] ≡ [B1, B1]. It is known [1, 2] that for any operator A = X + ZY , where X,Y,Z ∈ [B1] and Z is an involutive operator, Z2 = I, the Gohberg–Krupnik matrix equality is fulfilled: H [ A 0 0 A1 ] H?1 = D , where A1 is an additional associated operator, A1 = X ? ZY , and H = 1√ 2 [ I I Z ?Z ] , H?1 = 1√ 2 [ I Z I ?Z ] , D = [ X ZY Z Y ZXZ ] . 550 A. Karelin Comp.an.op.th. We denote the Cauchy singular integral operator along a contour Γ by (SΓ?)(t) = 1 πi ∫ Γ ?(τ) τ ? tdτ and the identity operator on Γ by (IΓ?)(t) = ?(t). Suppose that X = aIΓ + cSΓ , Y = (Zb)IΓ + (Zd)S