Some remarks on the method of sums (Q2702379)

The authors of this note present a short and self-contained proof of the method of sum due to \textit{G. Da Prato} and \textit{P. Grisvard} [J. Math. Pure Appl. IX. Ser. 54, 305-387 (1975; Zbl 0315.47009)] and \textit{G. Da Prato} [Analisi superiore, Lecture Notes, Course SNS Pisa 1983-1984].NEWLINENEWLINENEWLINELet \(X\) be a (complex) Banach space. A linear operator \(L:{\mathcal D}(L)\subset X\to X\) is nonnegative if \((-\infty, 0)\subset\rho(L)\) (the resolvent set of \(L\)) and NEWLINE\[NEWLINE\sup_{t>0} \|t(L+ tI)^{-1}\|< \infty.NEWLINE\]NEWLINE If \(L\) is a nonnegative operator on \(X\), then NEWLINE\[NEWLINE\phi_L\overset{\text{def}}= \sup\Biggl\{\phi\in [0,\pi]\mid \sup_{\substack{|\text{arg}(\lambda)|\leq\phi\\ \lambda\neq 0}} \|\lambda(L+ \lambda I)^{-1}\|< \infty\Biggr\},NEWLINE\]NEWLINE One usually says that \(\omega_L\overset{\text{def}} =\pi- \phi_L\) is the spectral angle of \(L\). They reproduce a proof of the theorem due to Da Prato and Griavard such as:NEWLINENEWLINENEWLINETheorem. Let \(X\) be a (complex) Banach space and assume thatNEWLINENEWLINENEWLINE(i) \(A\) and \(B\) are two linear operators on \(X\) with domains \({\mathcal D}(A)\) and \({\mathcal D}(B)\), respectively, and there are numbers \(\alpha\) and \(\beta\) in the resolvent sets \(\rho(A)\) and \(\rho(B)\) of \(A\) and \(B\), respectively, such that NEWLINE\[NEWLINE(A-\alpha I)^{-1}(B-\beta I)^{-1}= (B-\beta I)^{-1}(A-\alpha I)^{-1}.NEWLINE\]NEWLINE (ii) \(A\) and \(B\) are nonnegative operators on \(X\) and NEWLINE\[NEWLINE\phi_A+ \phi_B>\pi\text{ or, equivalently, }\omega_A+ \omega_B< \pi.NEWLINE\]NEWLINE (iii) \(0\in \rho(A)\cup\rho(B)\), i.e., at least one of the operators \(A\) and \(B\) is invertible.NEWLINENEWLINENEWLINEThen the following statements hold true:NEWLINENEWLINENEWLINE(a) There is a bounded linear operator \(S: X\to X\) such that NEWLINE\[NEWLINES= {1\over 2\pi i} \int_\gamma(A+ zI)^{-1}(B- zI)^{-1} dz,NEWLINE\]NEWLINE where \(\gamma\) is a path in \(\rho(-A)\cap \rho(B)\) separating the spectra of \(-A\) and \(B\), and NEWLINE\[NEWLINE\begin{aligned} S+ BA^{-1}S &= A^{-1}\quad\text{if }A\text{ is invertible},\\ AB^{-1}S+ S &= B^{-1}\quad \text{if }B\text{ is invertible}.\end{aligned}NEWLINE\]NEWLINE (b) If \(y\in{\mathcal D}(A)\cap{\mathcal D}(B)\), then \(S(Ay+ By)= y\).NEWLINENEWLINENEWLINE(c) If \(Sx\in{\mathcal D}(A)\cup{\mathcal D}(B)\) for some \(x\in X\), then \(Sx\in{\mathcal D}(A)\cap{\mathcal D}(B)\) and \(ASx+ BSx= x\).NEWLINENEWLINENEWLINE(d) The operator \(A+B\) with domain \({\mathcal D}(A)\cap{\mathcal D}(B)\) is closable in \(X\) and if \({\mathcal D}(A)+{\mathcal D}(B)\) is dense in \(X\), then \(S= \overline{(A+B)}^{-1}\).NEWLINENEWLINENEWLINEThe above theorem has also more additional results for numerical estimates.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].