Complex sector and bounded analytic semigroup

For $\delta \in (0,\pi ]$ put ${\mathrm{\Sigma }}_{\delta }:=\{z\in \mathbb{C}\setminus \{0\}:\mathrm{\mid }\arg z\mathrm{\mid }<\delta \},$ the open sector of half-angle $\delta$ around the positive real axis (A complex domain is a nonempty connected open subset of $\mathbb{C}$; we use the principal argument in $(-\pi ,\pi ]$). Let $X$ be a Banach space over $\mathbb{C}$ (Banach space). A family $(T(z){)}_{z\in {\mathrm{\Sigma }}_{\delta }\cup \{0\}}\subseteq \mathcal{B}(X)$ (A bounded linear operator between normed spaces, The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators) is an analytic semigroup of angle $\delta \in (0,\pi \mathrm{/}2]$ if:

(i) $T(0)=I$ and $T(z_1+z_2)=T(z_1)T(z_2)$ for all $z_1,z_2\in {\mathrm{\Sigma }}_{\delta }$;

(ii) $z\mapsto T(z)$ is holomorphic on ${\mathrm{\Sigma }}_{\delta }$ in the operator norm: the difference quotients $h^{-1}(T(z+h)-T(z))$ converge in $\mathcal{B}(X)$ for every $z\in {\mathrm{\Sigma }}_{\delta }$;

(iii) ${\lim }_{{\mathrm{\Sigma }}_{{\delta }^{\mathrm{\prime }}}\ni z\to 0}T(z)x=x$ for every $x\in X$ and every $0<{\delta }^{\mathrm{\prime }}<\delta$.

It is a bounded analytic semigroup of angle $\delta$ if in addition

(iv) ${\sup }_{z\in {\mathrm{\Sigma }}_{{\delta }^{\mathrm{\prime }}}}\mathrm{\parallel }T(z)\mathrm{\parallel }<\mathrm{\infty }$ for every $0<{\delta }^{\mathrm{\prime }}<\delta$.

Its generator is the infinitesimal generator of the strongly continuous semigroup $(T(t){)}_{t\ge 0}$ (Strongly continuous semigroup, Infinitesimal generator of a C0-semigroup), and its angle is the supremum of the $\delta$ for which such a family exists and extends the given one.

Strong continuity is required only at the vertex and only in the strong operator topology; holomorphy is asserted on the open sector, not at $0$. For a real Banach space $X$ the definition is applied through a complexification (Canonical Banach complexification of a real Banach space, Real and complex scalar conventions for normed spaces).

Remarks

• The sector ${\mathrm{\Sigma }}_{\delta }\cup \{0\}$ is a convex cone for $\delta \le \pi \mathrm{/}2$, so $z_1+z_2$ stays in the index set in (i); the vertex is the only boundary point at which values are prescribed. Condition (iii) is an assumption on the approach to the vertex along every strictly smaller sector, and it implies that $(T(t){)}_{t\ge 0}$ is a strongly continuous semigroup, since $[0,\mathrm{\infty })\subseteq {\mathrm{\Sigma }}_{\delta }\cup \{0\}$.
• No norm continuity at $0$ is asserted: when the generator is unbounded the family is only strongly continuous there, as the companion counterexample records. Likewise $T$ is not required to be holomorphic at $0$; only the values $T(z)$ for $z$ in the open sector carry the holomorphy of (ii).
• Condition (iv) is a boundedness requirement on every strictly smaller sector and not on all of ${\mathrm{\Sigma }}_{\delta }$; this is the distinction between a bounded analytic semigroup and an analytic semigroup whose norm may blow up near the boundary of the sector.