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Continuous free convolution on a finite time interval

Statement

Assume Countable Choice. Let $E$ be a complex Banach space regarded as a real Banach space, let $0 < T < \infty$ , and let $W (t)$ be a strongly continuous group of bounded real-linear operators on $E$ . Suppose $M = \sup_{∣ t ∣ \leq T} ∥ W (t) ∥ < \infty$ .

All time measurability, almost-everywhere assertions and integrals use one-dimensional Lebesgue measure restricted to $[0, T]$ : explicitly, $A_{T} = {A \cap [0, T] : A \in L (R)}$ and $μ_{T} (B) = λ_{1} (B)$ for $B \in A_{T}$ . Write $d s = d μ_{T} (s)$ . Functions initially defined on $(0, T)$ are extended by zero at the two endpoints.

For $f \in L^{1} (0, T; E)$ , meaning a strongly measurable function with $∥ f ∥_{L^{1}} = \int_{0}^{T} ∥ f (s) ∥ d s < \infty$ , and $r \in [0, T]$ , put $D_{r} f (t) = \int_{r}^{t} W (t - s) f (s) d s$ , where the integrals are Bochner integrals in the real Banach space $E$ and $\int_{r}^{t} = - \int_{t}^{r}$ if $t < r$ . Then $D_{r} f \in C ([0, T]; E)$ , $D_{r} f (r) = 0$ , and $∥ D_{r} f ∥_{C} \leq M ∥ f ∥_{L^{1}}$ , where $∥ v ∥_{C} = \sup_{t \in [0, T]} ∥ v (t) ∥$ .

For every $x \in E$ , the orbit $t \mapsto W (t - r) x$ is continuous on $[0, T]$ with supremum norm at most $M ∥ x ∥$ . The free forcing operator is real-linear, and bounded real-linear maps commute with its defining integrals.

Statement references

Banach space

The Axiom of Countable Choice (AC_ω)

Strongly continuous semigroup

Strongly measurable Banach-valued function

Bochner integrability criterion

Bochner integral norm inequality

Linearity of the Bochner integral

Bochner dominated convergence theorem

Bounded linear maps commute with Bochner integration

Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume

A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included

Bochner-integrable function

For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε

Facts & Assumptions

Given: The group on the realification of $E$ , the bound $M$ , the base time $r \in [0, T]$ , an arbitrary $x \in E$ , and strongly measurable integrable-norm $f$ , with the Lebesgue time measure specified in the Statement. All operator linearity is over $R$ .

Goal: Establish the continuous free forcing operator and its norm for either time orientation.

[F1]

Strong continuity and the group law have the meaning of Strongly continuous semigroup on the real Banach space of Banach space. Strong measurability is the simple-approximation convention Strongly measurable Banach-valued function.

[F2]

Bochner integrability and the integral are defined by norm limits of integrable-simple approximations in Bochner-integrable function, including restriction to measurable sets by indicators. Strong measurability plus integrable norm gives Bochner existence (Bochner integrability criterion); the norm inequality and real linearity are Bochner integral norm inequality and Linearity of the Bochner integral.

[F3]

Bochner dominated convergence under The Axiom of Countable Choice (AC_ω) is Bochner dominated convergence theorem. Bounded real-linear operators commute with integrals by Bounded linear maps commute with Bochner integration, applied over the real field.

[F4]

Under the assumed Countable Choice, Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume supplies the complete measure $(R, L (R), λ_{1})$ . Applying A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included with $n = 1$ makes every interval used here measurable with its length as measure; in particular each singleton is null. Each member of $A_{T}$ is Lebesgue measurable. Complements relative to $[0, T]$ and countable unions stay in $A_{T}$ , and the empty-set and disjoint-countable-additivity identities for $μ_{T}$ are inherited directly from $λ_{1}$ . Hence $([0, T], A_{T}, μ_{T})$ is a measure space; it is complete because every subset of a $μ_{T}$ -null set is Lebesgue measurable and contained in $[0, T]$ .

[F5]

Applying For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε in $R$ to $δ / T > 0$ shows that for every $δ > 0$ there is an integer $N \geq 1$ with $T / N < δ$ . Thus equal subdivisions of $[0, T]$ have arbitrarily small mesh.

Proof

Goal: Establish the continuous free forcing operator and its norm for either time orientation.

1.1F1F2F3F4F5givenconstruct

Fix $t \in [0, T]$ and let $f_{n}$ be measurable simple functions converging to $f$ outside a measurable null set. The group law and real linearity give, for $s, s^{'} \in [0, T]$ and $z \in E$ , $∥ W (t - s) z - W (t - s^{'}) z ∥ = ∥ W (t - s^{'}) (W (s^{'} - s) z - z) ∥ \leq M ∥ W (s^{'} - s) z - z ∥$ . For each $n \geq 1$ , strong continuity at zero, applied to the finitely many values $z$ of $f_{n}$ , gives a common $δ_{n} > 0$ such that $∥ W (h) z - z ∥ < 1 / (n (M + 1))$ for every such $z$ whenever $∣ h ∣ < δ_{n}$ . By [F5] choose $N_{n} \geq 1$ with $T / N_{n} < δ_{n}$ ; Countable Choice permits these choices simultaneously in $n$ . Partition $[0, T]$ into $N_{n}$ equal half-open intervals, including $T$ in the last interval, and let $q_{n} (s)$ be the left endpoint of its interval. Define $h_{n} (s) = W (t - q_{n} (s)) f_{n} (s)$ . This is a measurable simple function: its finitely many level pieces are intersections of subdivision intervals, measurable by [F4], with the measurable level sets of $f_{n}$ . The displayed group estimate gives $∥ h_{n} (s) - W (t - s) f_{n} (s) ∥ < 1 / n$ everywhere. Hence off the exceptional null set, $∥ h_{n} (s) - W (t - s) f (s) ∥ \leq 1 / n + M ∥ f_{n} (s) - f (s) ∥ \to 0$ . This proves strong measurability of $s \mapsto W (t - s) f (s)$ with an explicit uniform partition estimate. Its norm is at most $M ∥ f (s) ∥$ , so [F2] supplies its Bochner integral and its restriction to either oriented interval.

2.1F1F2F3F4step 1.1

Write the oriented integral over $[0, T]$ with scalar multiplier $a_{r, t} (s) = 1_{(r, t)} (s)$ when $t \geq r$ and $a_{r, t} (s) = - 1_{(t, r)} (s)$ otherwise. These indicators are measurable by [F4], and multiplication by them preserves the simple approximations in step 1.1. If $t_{n} \to t$ in $[0, T]$ , then $a_{r, t_{n}} (s) W (t_{n} - s) f (s) \to a_{r, t} (s) W (t - s) f (s)$ for almost every $s$ : the indicators are eventually constant away from $r, t$ , strong continuity handles the orbit, and those endpoints are null by [F4]. The dominating function is $M ∥ f (s) ∥$ . Dominated convergence proves continuity of $D_{r} f$ , including at $r$ , where its defining integral is zero. The norm inequality gives $∥ D_{r} f (t) ∥ \leq M \int_{\min (r, t)}^{\max (r, t)} ∥ f (s) ∥ d s$ .

3.1F1F2F3step 2.1∎

For the arbitrary vector $x \in E$ , the orbit assertion follows directly from [F1] and the assumed bound. The integral assertions about additivity, real scalar multiplication and bounded real-linear maps follow from [F2] and [F3], with the same oriented sign. Taking the supremum of the estimate in step 2.1 proves the stated $C$ -norm bound. The norm inequality also shows that changing $f$ on a measurable null set does not change $D_{r} f$ , so the forcing map is well defined on $L^{1} (0, T; E)$ .

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