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Restriction to a closed invariant subspace is a C0-semigroup and its generator is the part

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the Lebesgue-measure interfaces. Let $(T (t))_{t \geq 0}$ be a strongly continuous semigroup on a Banach space $X$ with generator $(A, D (A))$ , and let $Y \subseteq X$ be a closed linear subspace (Normed subspace, A closed subspace of a Banach space is Banach) such that $T (t) Y \subseteq Y$ for every $t \geq 0$ . Then the restrictions $T_{Y} (t) : = T (t) ∣_{Y}$ form a strongly continuous semigroup on the Banach space $Y$ , and its generator is the part $A_{Y}$ of $A$ in $Y$ : $D (A_{Y}) = {y \in D (A) \cap Y : A y \in Y}$ and $A_{Y} y = A y$ . In particular the generator of the restricted semigroup is the restriction of $A$ to that domain.

Facts & Assumptions

Given: Countable Choice; A strongly continuous semigroup $(T (t))_{t \geq 0}$ on a Banach space $X$ with generator $(A, D (A))$ (Strongly continuous semigroup, Infinitesimal generator of a C0-semigroup), and a closed linear subspace $Y \subseteq X$ with $T (t) Y \subseteq Y$ for every $t \geq 0$ .

[F1]

A closed linear subspace of a Banach space is a Banach space for the restricted norm (A closed subspace of a Banach space is Banach, Normed subspace), and convergence in the norm of $Y$ is the same as convergence in $X$ for vectors of $Y$ .

[F2]

The generator is defined by right difference quotients: $y \in D (A)$ exactly when $\frac{T (h) y - y}{h}$ converges as $h ↓ 0$ , and then the limit is $A y$ (Infinitesimal generator of a C0-semigroup).

[F3]

Time integrals of orbits lie in the generator domain: for $y \in X$ and $t > 0$ the Bochner integral $J_{t} y = \int_{0}^{t} T (s) y d s$ satisfies $J_{t} y \in D (A)$ and $A J_{t} y = T (t) y - y$ (Time integrals of semigroup orbits lie in the generator domain, Bochner-integrable function).

Proof

technique · direct: inheritance of the semigroup properties, then comparison of the two difference-quotient limits through the closed subspace

1.1F1

The restrictions $T_{Y} (t) : = T (t) ∣_{Y}$ are bounded linear maps of $Y$ into itself by hypothesis, with $T_{Y} (0) = I_{Y}$ and $T_{Y} (t + s) = T_{Y} (t) T_{Y} (s)$ inherited from $T$ . For each $y \in Y$ the orbit $t \mapsto T_{Y} (t) y$ is continuous into $Y$ , because it is continuous into $X$ and the norm of $Y$ is the restriction of the norm of $X$ by [F1]; hence $T_{Y}$ is a strongly continuous semigroup on the Banach space $Y$ .

1.2F1F2

Let $B$ denote the generator of $T_{Y}$ . If $y \in D (A) \cap Y$ and $A y \in Y$ , then $T (t) y \in Y$ for all $t$ , so the difference quotients $\frac{T_{Y} (h) y - y}{h} = \frac{T (h) y - y}{h}$ lie in $Y$ and converge in $X$ to $A y \in Y$ ; by [F1] they converge in $Y$ to $A y$ . Therefore $y \in D (B)$ and $B y = A y$ .

1.3F1F2

Conversely, if $y \in D (B)$ , then by definition $\frac{T_{Y} (h) y - y}{h} \to B y$ in $Y$ , hence also in $X$ by [F1]; the same vectors are the difference quotients of $T$ , so $y \in D (A)$ and $A y = B y$ . In particular $A y = B y \in Y$ , so $D (B) \subseteq {y \in D (A) \cap Y : A y \in Y}$ .

2.1F1F3step 1.2step 1.3∎

The two inclusions give $D (B) = {y \in D (A) \cap Y : A y \in Y}$ with $B y = A y$ , that is, the generator of $T_{Y}$ is the part $A_{Y}$ of $A$ in $Y$ ; for reference, this domain is dense in $Y$ , since for $y \in Y$ and $t > 0$ the integral $J_{t} y$ lies in $D (A) \cap Y$ with $A J_{t} y = T (t) y - y \in Y$ by [F3] and the $Y$ -valued Bochner integral stays in the closed subspace $Y$ , while $\frac{1}{t} J_{t} y \to y$ as $t ↓ 0$ .

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Sources

Klaus-Jochen Engel and Rainer Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194 (complete author-hosted monograph) (standard reference, not scraped)
Roland Schnaubelt, Evolution Equations, Karlsruhe Institute of Technology (2023/24 course, complete lecture notes) (standard reference, not scraped)