Row compactness of diagonal operators

Let R be a seminormed ring, I a discrete index type and d : I →ᵇ R a bounded family. The diagonal operator ZeroAtInftyContinuousMap.diagonalCLM d of C₀(I, R) has matrix Matrix.diagonal d, so its i-th row norm is ‖d i‖ (ZeroAtInftyContinuousMap.rowNorm_diagonalCLM). Hence a diagonal operator is row compact exactly when its entries tend to zero along the cofinite filter (ZeroAtInftyContinuousMap.isRowCompact_diagonalCLM_iff). Over an ultrametric ring it is then completely continuous, and over a complete ultrametric field Serre's criterion makes complete continuity equivalent to the decay of the entries (ZeroAtInftyContinuousMap.isCompletelyContinuous_diagonalCLM_iff).

On the other hand, over a ring with the strong rank condition, a diagonal operator whose entries are units at infinitely many indices does not have finite rank (ZeroAtInftyContinuousMap.not_isFiniteRank_diagonalCLM, in TauCeti.Topology.ContinuousMap.ZeroAtInfty.Diagonal). For a multiplicative pseudo-uniformiser ϖ the diagonal operator of the family ϖ ^ n on C₀(ℕ, R) is therefore completely continuous without having finite rank, so over an ultrametric Tate ring with the strong rank condition complete continuity is strictly weaker than finite rank (ZeroAtInftyContinuousMap.exists_isCompletelyContinuous_not_isFiniteRank).

Main results

• ZeroAtInftyContinuousMap.rowNorm_diagonalCLM: the row norms are the norms of the entries.
• ZeroAtInftyContinuousMap.isRowCompact_diagonalCLM_iff: a diagonal operator is row compact exactly when its entries tend to zero.
• ZeroAtInftyContinuousMap.isCompletelyContinuous_diagonalCLM_iff: over a complete ultrametric field, the same holds for complete continuity.
• ZeroAtInftyContinuousMap.exists_isCompletelyContinuous_not_isFiniteRank: over an ultrametric Tate ring with the strong rank condition, C₀(ℕ, R) carries a completely continuous endomorphism of infinite rank.

References

• J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. IHÉS 12 (1962), 69–85, §§2–3.

@[simp]

theorem ZeroAtInftyContinuousMap.rowNorm_diagonalCLM {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [SeminormedRing R] [DecidableEq I] (d : BoundedContinuousFunction I R) (i : I) : (diagonalCLM d).rowNorm i = ‖d i‖

The i-th row norm of the diagonal operator of d is ‖d i‖.

theorem ZeroAtInftyContinuousMap.isRowCompact_diagonalCLM_iff {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [SeminormedRing R] [DecidableEq I] {d : BoundedContinuousFunction I R} : (diagonalCLM d).IsRowCompact ↔ Filter.Tendsto (⇑d) Filter.cofinite (nhds 0)

Diagonal operators: the diagonal operator of d is row compact exactly when the entries d i tend to zero along the cofinite filter.

theorem ZeroAtInftyContinuousMap.isRowCompact_diagonalCLM {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [SeminormedRing R] [DecidableEq I] {d : BoundedContinuousFunction I R} (hd : Filter.Tendsto (⇑d) Filter.cofinite (nhds 0)) : (diagonalCLM d).IsRowCompact

A diagonal operator whose entries tend to zero is row compact.

theorem ZeroAtInftyContinuousMap.isCompletelyContinuous_diagonalCLM {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [NormedRing R] [IsUltrametricDist R] {d : BoundedContinuousFunction I R} (hd : Filter.Tendsto (⇑d) Filter.cofinite (nhds 0)) : (diagonalCLM d).IsCompletelyContinuous

Over an ultrametric ring, a diagonal operator whose entries tend to zero is completely continuous.

theorem ZeroAtInftyContinuousMap.exists_isCompletelyContinuous_not_isFiniteRank {R : Type u_2} [NormedRing R] [IsUltrametricDist R] [TauCeti.NormedRing.IsTate R] [StrongRankCondition R] : ∃ (u : ZeroAtInftyContinuousMap ℕ R →L[R] ZeroAtInftyContinuousMap ℕ R), u.IsCompletelyContinuous ∧ ¬(↑u).IsFiniteRank

Complete continuity strictly exceeds finite rank: over an ultrametric Tate ring with the strong rank condition, for instance a nonzero commutative one, the diagonal operator of the powers ϖ ^ n of a multiplicative pseudo-uniformiser is a completely continuous endomorphism of C₀(ℕ, R) that does not have finite rank.

theorem ZeroAtInftyContinuousMap.isCompletelyContinuous_diagonalCLM_iff {I : Type u_1} [TopologicalSpace I] [DiscreteTopology I] {K : Type u_3} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {d : BoundedContinuousFunction I K} : (diagonalCLM d).IsCompletelyContinuous ↔ Filter.Tendsto (⇑d) Filter.cofinite (nhds 0)

Serre's criterion for diagonal operators: over a complete ultrametric nontrivially normed field, a diagonal operator is completely continuous exactly when its entries tend to zero.