Pietsch domination

Let 1 ≤ p < ∞ and let B be the closed unit ball of the dual of E, carrying the weak-star topology, in which it is compact (Banach–Alaoglu). The Pietsch domination theorem says that an operator T : E →L[𝕜] F satisfies πₚ(T) ≤ C exactly when there is a regular Borel probability measure ν on B with ‖T x‖ ≤ C (∫ |f x|ᵖ dν(f))^(1/p) for all x : E.

The dominated estimate gives the bound on πₚ(T) by integrating the pointwise inequality ∑ᵢ |f (xᵢ)|ᵖ ≤ (weak p-norm of x)ᵖ against ν. Conversely, the functions f ↦ ∑ᵢ (‖T xᵢ‖ᵖ - Cᵖ |f (xᵢ)|ᵖ) form a convex cone in C(B, ℝ) that the summing inequality keeps disjoint from the open cone of functions bounded below by a positive constant. A Hahn–Banach separation of these two cones produces a positive linear functional on C(B, ℝ), and the Riesz–Markov–Kakutani theorem represents it, after normalization, by the measure ν.

The measurable structure on B is the subtype structure of a Borel structure on the weak dual, so ν is a Borel measure for the weak-star topology.

Main results

• ContinuousLinearMap.summingENorm_le_ofReal_of_norm_le_integral: a dominating probability measure bounds the p-summing gauge.
• ContinuousLinearMap.exists_isProbabilityMeasure_of_summingENorm_le_ofReal: an operator with πₚ(T) ≤ C has a regular dominating probability measure.
• ContinuousLinearMap.exists_isProbabilityMeasure_of_summingENorm_ne_top: an absolutely p-summing operator is dominated with the constant πₚ(T).
• ContinuousLinearMap.summingENorm_le_ofReal_iff_exists_isProbabilityMeasure: the Pietsch domination theorem.

References

• F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Theorem 8.2.8.
• J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Theorem 2.12.

theorem ContinuousLinearMap.summingENorm_le_ofReal_of_norm_le_integral {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] {p : ENNReal} [Fact (1 ≤ p)] [MeasurableSpace (WeakDual 𝕜 E)] [OpensMeasurableSpace (WeakDual 𝕜 E)] (hp : p ≠ ⊤) {T : E →L[𝕜] F} {C : ℝ} (hC : 0 ≤ C) (ν : MeasureTheory.Measure ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)) [MeasureTheory.IsProbabilityMeasure ν] (hT : ∀ (x : E), ‖T x‖ ≤ C * (∫ (f : ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)), ‖↑f x‖ ^ p.toReal ∂ν) ^ (1 / p.toReal)) : summingENorm p T ≤ ENNReal.ofReal C

Pietsch domination, the easy direction: if ‖T x‖ ≤ C (∫ |f x|ᵖ dν(f))^(1/p) for a probability measure ν on the dual unit ball, then πₚ(T) ≤ C.

theorem ContinuousLinearMap.exists_isProbabilityMeasure_of_summingENorm_le_ofReal {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] {p : ENNReal} [Fact (1 ≤ p)] [MeasurableSpace (WeakDual 𝕜 E)] [BorelSpace (WeakDual 𝕜 E)] (hp : p ≠ ⊤) {T : E →L[𝕜] F} {C : ℝ} (hC : 0 ≤ C) (hT : summingENorm p T ≤ ENNReal.ofReal C) : ∃ (ν : MeasureTheory.Measure ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)), MeasureTheory.IsProbabilityMeasure ν ∧ ν.Regular ∧ ∀ (x : E), ‖T x‖ ≤ C * (∫ (f : ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)), ‖↑f x‖ ^ p.toReal ∂ν) ^ (1 / p.toReal)

Pietsch domination, the main direction: if πₚ(T) ≤ C with 1 ≤ p < ∞, there is a regular Borel probability measure ν on the weak-star compact dual unit ball such that ‖T x‖ ≤ C (∫ |f x|ᵖ dν(f))^(1/p) for all x.

theorem ContinuousLinearMap.exists_isProbabilityMeasure_of_summingENorm_ne_top {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] {p : ENNReal} [Fact (1 ≤ p)] [MeasurableSpace (WeakDual 𝕜 E)] [BorelSpace (WeakDual 𝕜 E)] (hp : p ≠ ⊤) {T : E →L[𝕜] F} (hT : summingENorm p T ≠ ⊤) : ∃ (ν : MeasureTheory.Measure ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)), MeasureTheory.IsProbabilityMeasure ν ∧ ν.Regular ∧ ∀ (x : E), ‖T x‖ ≤ (summingENorm p T).toReal * (∫ (f : ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)), ‖↑f x‖ ^ p.toReal ∂ν) ^ (1 / p.toReal)

An absolutely p-summing operator, 1 ≤ p < ∞, is dominated by a regular Borel probability measure on the dual unit ball with the constant πₚ(T).

theorem ContinuousLinearMap.summingENorm_le_ofReal_iff_exists_isProbabilityMeasure {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] {p : ENNReal} [Fact (1 ≤ p)] [MeasurableSpace (WeakDual 𝕜 E)] [BorelSpace (WeakDual 𝕜 E)] (hp : p ≠ ⊤) {T : E →L[𝕜] F} {C : ℝ} (hC : 0 ≤ C) : summingENorm p T ≤ ENNReal.ofReal C ↔ ∃ (ν : MeasureTheory.Measure ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)), MeasureTheory.IsProbabilityMeasure ν ∧ ν.Regular ∧ ∀ (x : E), ‖T x‖ ≤ C * (∫ (f : ↑(⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 1)), ‖↑f x‖ ^ p.toReal ∂ν) ^ (1 / p.toReal)

Pietsch domination theorem. For 1 ≤ p < ∞ and C ≥ 0, an operator satisfies πₚ(T) ≤ C if and only if there is a regular Borel probability measure ν on the weak-star compact dual unit ball with ‖T x‖ ≤ C (∫ |f x|ᵖ dν(f))^(1/p) for all x.