Row-compact operators between model spaces #
Let R be a normed ring and I, J discrete index types. A continuous linear map
u : C₀(J, R) →L[R] C₀(I, R) has the matrix aᵢⱼ = u.matrixCoeff i j, whose rows are indexed by
output coordinates. Its row norms are the suprema rᵢ(u) = ⨆ j, ‖aᵢⱼ‖
(ContinuousLinearMap.rowNorm), and u is row compact (ContinuousLinearMap.IsRowCompact)
if its entries are uniformly bounded and its row norms tend to zero along the cofinite filter.
The bound on the entries keeps the real suprema meaningful; it is automatic over Tate scalars.
Row compactness is the matrix condition of Serre's criterion for complete continuity. For an
endomorphism u of C₀(I, R) over a complete commutative ultrametric normed field, it is also
the condition under which Serre's Fredholm determinant det (1 - T u) is an entire power series;
that determinant is not developed here.
Every column of the matrix of a continuous linear map tends to zero, but its rows need not: the
identity of C₀(I, R) has all row norms one, so it is not row compact on an infinite index type
(ContinuousLinearMap.not_isRowCompact_id).
Over ultrametric Tate scalars the operator norm is the supremum of the row norms
(ContinuousLinearMap.ringOpNorm_eq_iSup_rowNorm), and row compactness says exactly that the
output truncations π_S ∘ u approximate u in operator norm as the finite set S of output
coordinates grows (ContinuousLinearMap.isRowCompact_iff_tendsto_ringOpNorm_sub_indicatorCLM_comp).
Here π_S = ZeroAtInftyContinuousMap.indicatorCLM R S is the coordinate projection. These
truncations have finite rank, so every row-compact map is completely continuous
(ContinuousLinearMap.IsRowCompact.isCompletelyContinuous), over any ultrametric normed ring and
with no Noetherian hypothesis. The input truncations u ∘ π_S need not converge, because a
row-compact map of rank one can have a row of entries of norm one.
Row-compact maps form a closed two-sided operator ideal over ultrametric Tate scalars: they are
stable under sums and scalar multiples, under composition on either side with continuous linear
maps (ContinuousLinearMap.IsRowCompact.comp_clm, ContinuousLinearMap.IsRowCompact.clm_comp),
and under limits in operator norm (ContinuousLinearMap.isRowCompact_of_tendsto). In particular
a polynomial with zero constant coefficient takes row-compact endomorphisms to row-compact
endomorphisms (ContinuousLinearMap.IsRowCompact.aeval).
Main definitions #
ContinuousLinearMap.rowNorm: the row norms of the matrix of a map between model spaces.ContinuousLinearMap.IsRowCompact: bounded entries and row norms tending to zero.
Main results #
ContinuousLinearMap.isRowCompact_of_norm_matrixCoeff_le: a null row majorant of the entries gives row compactness.ContinuousLinearMap.isRowCompact_of_apply_single_eq_sum: columns that are uniformly bounded combinations of finitely many vectors give a row-compact map.ContinuousLinearMap.ringOpNorm_eq_iSup_rowNorm:‖u‖ = ⨆ i, rᵢ(u)over ultrametric Tate scalars.ContinuousLinearMap.isRowCompact_iff_tendsto_ringOpNorm_sub_indicatorCLM_comp: row compactness is norm convergence of the output truncations.ContinuousLinearMap.IsRowCompact.tendsto_ringOpNorm_indicatorCLM_comp_sub: the output truncations of a row-compact map converge to it in operator norm.ContinuousLinearMap.IsRowCompact.isCompletelyContinuous: row-compact maps are completely continuous.
References #
- J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. IHÉS 12 (1962), 69–85, §3.
- C. Johansson and J. Newton, Extended eigenvarieties for overconvergent cohomology, arXiv:1604.07739, §2.1.
The row norms of a continuous linear map u : C₀(J, R) →L[R] C₀(I, R): the norm of row
i of its matrix is ⨆ j, ‖u.matrixCoeff i j‖. As a real supremum this is only meaningful for a
bounded row, which is automatic over Tate scalars (ContinuousLinearMap.rowNorm_le_ringOpNorm).
Instances For
A bound on every entry of a row bounds its row norm.
In a bounded row, every entry is bounded by the row norm.
A continuous linear map between model spaces is row compact if the entries of its matrix
are uniformly bounded and its row norms tend to zero along the cofinite filter of output indices.
The bound is automatic over Tate scalars (ContinuousLinearMap.isRowCompact_iff_tendsto_rowNorm);
it is part of the definition so that the real suprema defining the row norms are meaningful.
Instances For
The entries of a row-compact map are bounded by its row norms.
Row majorants: if every entry of row i is bounded by r i, where r tends to zero
along the cofinite filter, then u is row compact.
Bounded finite expansions of the columns give row decay. If every column
u (single j 1) of u is a combination ∑ k, a k • w k of a fixed finite family w with
coefficients of norm at most B, then u is row compact: row i is bounded by
∑ k, B * ‖w k i‖.
Scalar multiples of row-compact maps are row compact.
Over Tate scalars, every row norm is bounded by the operator norm.
Over Tate scalars, every entry is bounded by its row norm.
A row norm is bounded by the operator-norm distance to another map plus its row norm.
Over Tate scalars, a row norm is at most C ≥ 0 exactly when every entry of the row is.
Over Tate scalars, row compactness only asks the row norms to tend to zero.
Over Tate scalars, every continuous linear map between model spaces on a finite output index type is row compact.
Closedness: over Tate scalars, a limit in operator norm of row-compact maps is row compact.
The identity is not row compact on an infinite index type over a nonzero norm-one ring: every row of its matrix has norm one.
On an infinite model space a scalar multiple of the identity is row compact exactly when its scalar is zero. No invertibility assumption on the scalar is needed.
The sup-row norm: over ultrametric Tate scalars, the operator norm of a map between model spaces is the supremum of its row norms.
Over ultrametric Tate scalars, the i-th coordinate of u x is bounded by the row norm
rᵢ(u) times ‖x‖.
The output truncation π_S ∘ u keeps the rows of the matrix of u indexed by S and
replaces the others by zero.
The row norms of the output truncation π_S ∘ u are those of u on S and zero
elsewhere.
Removing the output truncation π_S ∘ u from u leaves the truncation to the complement
of S.
Over Tate scalars, the output truncation π_S ∘ u to a finite set S is row compact: its
rows outside S vanish.
Over an ultrametric ring, if the entries of u in the rows outside S have norm at most
C, the remainder u - π_S ∘ u of the output truncation satisfies
‖(u - π_S ∘ u) x‖ ≤ C * ‖x‖.
Row-compact maps are completely continuous, over any ultrametric normed ring: they are uniformly approximated by their output truncations onto finite sets, which have finite rank.
Row-compact operators have output truncations π_S ∘ u converging to u in operator norm.
The output truncations π_S ∘ u of a row-compact u converge to u in operator norm as the
finite set S grows.
Row compactness is norm convergence of output truncations: over ultrametric Tate scalars,
u is row compact exactly when ‖u - π_S ∘ u‖ → 0 as the finite set S of output coordinates
grows.
Over ultrametric Tate scalars, precomposing a row-compact map with a continuous linear map
gives a row-compact map: row i of u ∘ w has norm at most rᵢ(u) * ‖w‖.
Over ultrametric Tate scalars, postcomposing a row-compact map with a continuous linear map
gives a row-compact map. The finitely many large rows of u are absorbed by the null columns of
the matrix of w.
A polynomial with zero constant coefficient takes row-compact operators to row-compact operators, over ultrametric Tate scalars.
A norm-convergent power series with zero constant term in a row-compact operator has a row-compact value. Entirety of the scalar series is not required.