Sectorial operator with the semigroup sign convention

Let $X$ be a Banach space over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ (Banach space, Real and complex scalar conventions for normed spaces) and let $A:D(A)\subseteq X\to X$ be a closed densely defined linear operator with resolvent $R(\lambda ,A)=(\lambda I-A{)}^{-1}$ (Resolvent and spectrum of a closed operator on a Banach space, Densely defined, closed and closable operators, and cores, Unbounded linear operators: domain, graph and extension).

For a real $X$, all complex resolvents below are those of the closed complexified operator $A_{\mathbb{C}}(x,y)=(Ax,Ay)$ on $D(A)\times D(A)$ in the canonical complexification (Canonical Banach complexification of a real Banach space). Closedness and density follow coordinatewise, since its norm is equivalent to the product norm.

For $\delta \in (0,\pi \mathrm{/}2]$ and $\omega \in \mathbb{R}$, $A$ (or the pair $(A,\omega )$) is sectorial of angle $\delta$ with vertex $\omega$ in the $e^{tA}$ convention if the open sector $\omega +{\mathrm{\Sigma }}_{\pi \mathrm{/}2+\delta }$ (Complex sector and bounded analytic semigroup) is contained in the resolvent set $\rho (A)$, and for every $\epsilon \in (0,\delta )$ there is a constant $M_{\epsilon }\ge 1$ with $\mathrm{\parallel }R(\lambda ,A)\mathrm{\parallel }\le \frac{M_{\epsilon }}{\mathrm{\mid }\lambda -\omega \mathrm{\mid }}\text{for all }\lambda \in \omega +{\mathrm{\Sigma }}_{\pi \mathrm{/}2+\delta -\epsilon }\mathrm{.}$

The sign dictionary

The definition is equivalent to the pair of statements that the spectrum of $A$ is contained in the complementary closed left sector $\sigma (A)\subseteq \omega +(\{\lambda \mathrm{\ne }0:\mathrm{\mid }\arg (-\lambda )\mathrm{\mid }\le \pi \mathrm{/}2-\delta \}\cup \{0\})$ and that the stated $M_{\epsilon }\mathrm{/}\mathrm{\mid }\lambda -\omega \mathrm{\mid }$ bound holds on the right-opening sector. Writing $B:=-A+\omega I$ one has $(\mu I-B)=(\mu -\omega )I+A=-((\omega -\mu )I-A),$ hence $R(\mu ,B)=-R(\omega -\mu ,A)$; substituting $\lambda =\omega -\mu$, the resolvent bound of $A$ on $\omega +{\mathrm{\Sigma }}_{\pi \mathrm{/}2+\delta -\epsilon }$ becomes the bound $\mathrm{\parallel }R(\mu ,B)\mathrm{\parallel }\le M_{\epsilon }\mathrm{/}\mathrm{\mid }\mu \mathrm{\mid }$ on the reflected left-opening sector $-{\mathrm{\Sigma }}_{\pi \mathrm{/}2+\delta -\epsilon }$ for $\mu$, so that sector lies in $\rho (B)$ and the spectrum of $B$ lies in the closed sector $\{\mu \mathrm{\ne }0:\mathrm{\mid }\arg \mu \mathrm{\mid }\le \pi \mathrm{/}2-\delta \}\cup \{0\}$.

This dictionary is why every theorem on this page states its convention: a source that calls $A$ "sectorial" for the opposite operator $-A$, or that writes $e^{-t(-A)}$, is using the Pazy-Lunardi sign and its sector and angle must be reflected before transfer. The free use of laplace-transform-shaped formulas below is always in the $e^{tA}$ convention fixed here: the resolvent sector of $A$ opens around the positive real direction, the spectral sector lies to the left, and positive time corresponds to an integral of $e^{\lambda z}R(\lambda ,A)$.