Real Banach spaces require complexification for analyticity

Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) for the cited integral and semigroup suppliers.

Given a real Banach space $X$ and a strongly continuous semigroup $(T(t){)}_{t\ge 0}$ of bounded real-linear operators, first complexify each real-time operator to $T_{\mathbb{C}}(t):=(T(t){)}_{\mathbb{C}}$ on $X_{\mathbb{C}}=X\times X$ with the rotation-supremum norm (Canonical Banach complexification of a real Banach space). The real semigroup is analytic of angle $\delta$ when this complexified real-time semigroup admits an analytic extension ${\tilde{T}}_{\mathbb{C}}(z)$ to ${\mathrm{\Sigma }}_{\delta }\cup \{0\}$ on $X_{\mathbb{C}}$, agreeing with $T_{\mathbb{C}}(t)$ for $t\ge 0$; the real-time operators preserve the embedded copy $X\times \{0\}$; preservation at nonreal times is not guaranteed, but can occur (the identity semigroup preserves it at every complex time) (Complex sector and bounded analytic semigroup). It is bounded analytic when the extension is bounded on every strictly smaller sector. A holomorphic map is complex-time: a real-linear family defined only for real $t$ is not itself a map on a complex sector.

For a bounded real-linear operator its spectrum and resolvent are computed on its complexification using Resolvent and spectrum of a closed operator on a Banach space; no spectral-radius assertion is used here. For an unbounded real generator $A$, complexify its domain and action, $A_{\mathbb{C}}(x,y)=(Ax,Ay)$ on $D(A)\times D(A)\subset X_{\mathbb{C}}$, then impose the closed-unbounded resolvent and sectorial conditions on $A_{\mathbb{C}}$ using Resolvent and spectrum of a closed operator on a Banach space and Sectorial operator with the semigroup sign convention. The sectorial-generation theorem Sectorial resolvent characterisation of bounded analytic semigroups is applied to that complexified operator. The heat equation on real $L^2$ is recovered by restricting $T_{\mathbb{C}}(t)$ to the real summand for real $t\ge 0$; its holomorphic extension is on the complexification, not a real-valued map at nonreal times.