Summary
In Section 4.1, we characterize, in terms of the estimates of <m:math display='block'> <m:mrow> <m:msup> <m:mi>λ</m:mi> <m:mrow> <m:mi>n</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow> </m:msup> <m:msub> <m:mi>R</m:mi> <m:mi>λ</m:mi> </m:msub> <m:mo>,</m:mo><m:msup> <m:mi>λ</m:mi> <m:mrow> <m:mi>k</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow> </m:msup> <m:msub> <m:mi>A</m:mi> <m:mi>k</m:mi> </m:msub> <m:msub> <m:mi>R</m:mi> <m:mi>λ</m:mi> </m:msub> <m:mo>,</m:mo><m:msup> <m:mi>λ</m:mi> <m:mrow> <m:mi>k</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow> </m:msup> <m:mover accent='true'> <m:mrow> <m:msub> <m:mi>R</m:mi> <m:mi>λ</m:mi> </m:msub> <m:msub> <m:mi>A</m:mi> <m:mi>k</m:mi> </m:msub> </m:mrow> <m:mo stretchy='true'>¯</m:mo> </m:mover> <m:mrow><m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn><m:mo>≤</m:mo><m:mi>k</m:mi><m:mo>≤</m:mo><m:mi>n</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow> <m:mo>)</m:mo></m:mrow></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\lambda ^{n - 1}}{R_\lambda },{\lambda ^{k - 1}}{A_k}{R_\lambda },{\lambda ^{k - 1}}\overline {{R_\lambda }{A_k}} \left( {1 \leqslant k \leqslant n - 1} \right)$$ , those (ACP n ) whose propagators can be extended analytically to the sector Σ θ (for a fixed <m:math display='block'> <m:mrow> <m:mi>θ</m:mi><m:mo>∈</m:mo><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn><m:mo>,</m:mo><m:mfrac> <m:mi>π</m:mi> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>]</m:mo></m:mrow></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\theta \in \left( {0,\frac{\pi }{2}} \right]$$ ) satisfying appropriate conditions there; such behavior of (ACP n ) is called analytic wellposedness (in Σ θ ), which will be made precise in Definition 1.2. We also treat perturbation problems about analytic wellposedness in this section. A new type of perturbation operators is introduced, besides that given in Section 2.4.