Kenig, C.; Meyer, Y. Kato’s square roots of accretive operators and Cauchy kernels on Lipschitz curves are the same. (English) Zbl 0641.47039 Recent progress in Fourier analysis, Proc. Semin., El Escorial/Spain 1983, North-Holland Math. Stud. 111, 123-143 (1985). [For the entire collection see Zbl 0581.00009.] Kato’s definition of the square root \(\sqrt{T}\) of an m-accretive operator T is slightly generalized and used in the proof of the following theorem: Let A and B be pointwise multiplication operators by \(a\in L^{\infty}\) and \(b\in L^{\infty}\), respectively, let \(D=-id/dx\) and let \(T=BDAD\). Then \((1+\epsilon T)^{-1}\) is bounded on L 2 for every \(\epsilon\geq 0\), the norms of these operators are bounded independently of \(\epsilon\) and the domain of \(\sqrt{T}\) is the Sobolev space H 1. Moreover, \(\sqrt{T}=J(A,B)D\), where J(A,B):L \(2\to L\) 2 is an isomorphism. The second main result is concerned with the Cauchy operator \(C_{\Phi}\) whose kernel is \((i\pi)^{-1}PV[z(y)-z(x)]^{-1}\), where \(z(x)=x+i\Phi (x)\) and \(\Phi\) denotes a Lipschitz function. It is then proved that \(C_{\Phi}=J(A,A)\), where J(A,A) is defined as in the preceding theorem with \(a(x)=[1+i\Phi '(x)]^{-1}\). This result shows that the boundedness of the Cauchy operator is a special case of a more general theorem concerning square roots of second-order differential operators. Cited in 1 ReviewCited in 10 Documents MSC: 47B44 Linear accretive operators, dissipative operators, etc. 47E05 General theory of ordinary differential operators 47A60 Functional calculus for linear operators 47Gxx Integral, integro-differential, and pseudodifferential operators Keywords:square root of an m-accretive operator; pointwise multiplication operators; Cauchy operator; Lipschitz function; square roots of second- order differential operators Citations:Zbl 0581.00009 PDFBibTeX XML