The heat operator family is an analytic semigroup in the later abstract language

Orientation only. For $1\le p<\mathrm{\infty }$, the family $z\mapsto H_z$ constructed explicitly in Complex-time heat operators form a bounded holomorphic semigroup is the concrete Gaussian instance of what the library's later abstract treatment of analytic semigroups will call a bounded holomorphic (analytic) $C_0$-semigroup on a sector: bounded on every proper subsector, strongly continuous on the positive real axis, and multiplicative in the sector. Concretely, the kernel and its estimates are those of The complex-time heat kernel on a proper sector; the sectorial bound $\mathrm{\parallel }{H}_z{\mathrm{\parallel }}_{L^p\to L^p}\le (\cos \sigma {)}^{-n\mathrm{/}2}$ on $\mathrm{\mid }\arg z\mathrm{\mid }\le \sigma <\theta$ and the law $H_z{H}_w=H_{z+w}$ are clauses (i) and (ii) of that theorem, its clause (iii) gives the holomorphy in operator norm, and the strong continuity at the vertex along the positive axis, for $1\le p<\mathrm{\infty }$, is that of the heat flow of The heat evolution $H_t$ of initial data supplied by The heat Cauchy problem for $L^p$ data.

This page does not use that abstract notion as a premise, does not identify the generator $\mathrm{\Delta }$ with an unbounded operator domain, and makes no claim about maximal regularity, resolvent sectors, or the Hille–Yosida representation in the abstract language. The later page is responsible for the abstract definition and for the generator theory; here only the explicit kernel family of The complex-time heat kernel on a proper sector and its estimates are used.

For $p=\mathrm{\infty }$ the same family is bounded and operator-norm holomorphic at positive complex times, but is not a $C_0$-semigroup on all of $L^{\mathrm{\infty }}$; the vertex continuity assertion above is restricted to finite $p$. Indeed, for $f={\mathbf{1}}_{[0,\mathrm{\infty })}$ on $\mathbb{R}$, evenness and unit mass give $H_{t}f(0)=1\mathrm{/}2$. Continuity of $H_{t}f$ makes $\mathrm{\mid }{H}_{t}f(x)-1\mathrm{\mid }>\eta$ on a positive-length interval $0<x<\delta$ for every $\eta <1\mathrm{/}2$; hence $\mathrm{\parallel }{H}_{t}f-f{\mathrm{\parallel }}_{\mathrm{\infty }}\ge 1\mathrm{/}2$ for every $t>0$.

For finite $p$, continuity at the vertex also holds within each proper subsector. With $\epsilon >0$ real and $\mathrm{\mid }\arg z\mathrm{\mid }\le \sigma$, the semigroup law gives $\mathrm{\parallel }{H}_{z}f-f{\mathrm{\parallel }}_p\le ((\cos \sigma {)}^{-n\mathrm{/}2}+1)\mathrm{\parallel }f-H_{\epsilon }f{\mathrm{\parallel }}_p+\mathrm{\parallel }{H}_{z+\epsilon }f-H_{\epsilon }f{\mathrm{\parallel }}_p\mathrm{.}$ For fixed $\epsilon$, the last term tends to zero as $z\to 0$ by positive-parameter operator holomorphy; then $\epsilon \downarrow 0$ controls the first term by the finite-$p$ real-time continuity cited above. This supplies the vertex continuity of the analytic $C_0$ terminology.