The Riesz projector of a row-compact operator at a good zero #
Let R be a complete ultrametric normed commutative ring with Tate scalars, I a discrete index
type, and u a row-compact endomorphism of the model space C₀(I, R) with Fredholm determinant
H_u(T) = det (1 - T u), the principal-minor series u.matrixCoeff.charPowerSeries. Suppose
that a is a good zero of order h of H_u: the Hasse derivatives ΔʲH_u(a) vanish for
j < h and c = ΔʰH_u(a) is a unit of R. Put C = 1 - a • u.
A Riesz projection of u at a of order h (ContinuousLinearMap.IsRieszProjection) is an
idempotent E, commuting with u and an operator-norm limit of polynomials in u, with
C ^ h * E = 0, such that C is invertible on the complementary summand: there is w, again a
limit of polynomials in u, with C w = w C = 1 - E and E w = w E = 0. Thus E splits
C₀(I, R) into the generalised eigenspace range E = ker C ^ h
(ContinuousLinearMap.IsRieszProjection.range_eq_ker), on which C is nilpotent, and a stable
complement range (1 - E) on which C is invertible. Such a splitting is unique
(ContinuousLinearMap.IsRieszProjection.unique); at order 0 it is E = 0. The definition and
these algebraic consequences make sense for an endomorphism of any seminormed module with a
bounded scalar action, and are stated in that generality; only the construction below uses the
model space.
Riesz projections compress along continuous linear retracts r ∘ i = id of M through a normed
module N: if E is a Riesz projection of the extension i ∘ u ∘ r, then r ∘ E ∘ i is one of
u (ContinuousLinearMap.IsRieszProjection.comp_comp_of_retract). The idempotent i ∘ r
commutes with the operator-norm closure of the polynomials in i ∘ u ∘ r, on which the
compression X ↦ r ∘ X ∘ i is multiplicative and continuous, so it carries the defining identities
and the closure condition to M.
The Riesz projector is constructed from the Hasse derivatives of the Fredholm resolvent
V(T) = H_u(T) (1 - T u)⁻¹: with b = c⁻¹ • ΔʰV(a) the differentiated resolvent identity gives
C ^ (h + 1) b = C ^ h, and
E = 1 - (C b) ^ h (ContinuousLinearMap.rieszProjection)
is a Riesz projection, with w = b (C b) ^ h
(ContinuousLinearMap.IsRowCompact.isRieszProjection_rieszProjection). For h > 0 the operator
1 - C b = -c⁻¹ • u Δʰ⁻¹V(a) is row compact, so E is row compact
(ContinuousLinearMap.IsRowCompact.isRowCompact_rieszProjection). More generally, every Riesz
projection of a row-compact operator is row compact, including at order 0, since
E = a • u * (∑ i < h, C ^ i) * E (ContinuousLinearMap.IsRieszProjection.isRowCompact).
Hence its range is a finitely generated projective R-module
(ContinuousLinearMap.IsRowCompact.finite_range_of_isIdempotentElem,
ContinuousLinearMap.IsRowCompact.projective_range_of_isIdempotentElem). The scalars may have
nilpotents and need not be Noetherian; over a ring the range need not be free.
Main definitions #
ContinuousLinearMap.IsRieszProjection: the finite spectral projector ofuataof orderh.ContinuousLinearMap.IsRieszProjection.inverse: the inverse ofCon the complementary summand, extended by zero.ContinuousLinearMap.rieszProjection: the explicit projector1 - (C b) ^ h.
Main results #
ContinuousLinearMap.IsRieszProjection.range_eq_ker: the range of a Riesz projection is the generalised eigenspaceker (1 - a • u) ^ h.ContinuousLinearMap.IsRieszProjection.maxGenEigenspace_eq_range: fora * μ = 1it is the maximal generalised eigenspace ofuatμ.ContinuousLinearMap.IsRieszProjection.unique: Riesz projections are unique.ContinuousLinearMap.IsRieszProjection.comp_comp_of_retract: Riesz projections compress along continuous linear retracts.ContinuousLinearMap.IsRowCompact.isRieszProjection_rieszProjection: at a good zero, the explicit projector is a Riesz projection.ContinuousLinearMap.IsRowCompact.isRowCompact_rieszProjection: it is row compact.ContinuousLinearMap.IsRieszProjection.isRowCompact: every Riesz projection of a row-compact operator is row compact.
References #
- J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. IHÉS 12 (1962), 69–85, §7.
- R. Coleman, p-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479, Appendix A, A4.
- C. Johansson and J. Newton, Extended eigenvarieties for overconvergent cohomology, arXiv:1604.07739, §2.2.
A Riesz projection of an endomorphism u of a seminormed R-module M at the scalar
a, of order h: an idempotent E commuting with u and lying in the operator-norm closure of
the polynomials in u, such that (1 - a • u) ^ h * E = 0 and 1 - a • u is invertible on the
complementary summand range (1 - E), with inverse again in the closure of the polynomials in
u. For a good zero a of order h of the Fredholm determinant of a row-compact u, the range
of E is the generalised eigenspace ker (1 - a • u) ^ h, and E is the finite spectral
projector onto it.
- isIdempotentElem : IsIdempotentElem E
A Riesz projection is idempotent.
- commute : Commute u E
A Riesz projection commutes with the operator.
A Riesz projection is an operator-norm limit of polynomials in the operator.
(1 - a • u) ^ hvanishes on the range of a Riesz projection.- exists_inverse : ∃ (w : M →L[R] M), (1 - a • u) * w = 1 - E ∧ w * (1 - a • u) = 1 - E ∧ E * w = 0 ∧ w * E = 0 ∧ w ∈ closure ↑R[u]
1 - a • uis invertible on the complementary summand, with inverse an operator-norm limit of polynomials in the operator.
Instances For
A Riesz projection of order 0 is zero.
A Riesz projection commutes with 1 - a • u.
A Riesz projection remains one when its nilpotence bound is increased.
The operator preserves the kernel of its Riesz projection.
The operator preserves the range of its Riesz projection.
The spectral operator preserves the kernel of its Riesz projection.
The spectral operator preserves the range of its Riesz projection.
The complementary inverse of a Riesz projection: the inverse of 1 - a • u on the
complementary summand range (1 - E), extended by zero on range E, chosen from
IsRieszProjection.exists_inverse.
Instances For
The complementary inverse is a right inverse of 1 - a • u up to 1 - E.
The complementary inverse is a left inverse of 1 - a • u up to 1 - E.
The complementary inverse takes values in the kernel of the Riesz projection.
The complementary inverse vanishes on the range of the Riesz projection.
The complementary inverse is an operator-norm limit of polynomials in the operator.
The complementary inverse is the unique right inverse of 1 - a • u up to 1 - E that
takes values in the kernel of the Riesz projection.
The complementary inverse preserves the kernel of the Riesz projection.
On the complementary summand, the complementary inverse is a left inverse of the spectral operator.
On the complementary summand, the complementary inverse is a right inverse of the spectral operator.
The range of a Riesz projection is the generalised eigenspace ker (1 - a • u) ^ h.
The range of a Riesz projection is the maximal generalised eigenspace. If a * μ = 1,
the range of a Riesz projection of u at a is the space of all vectors killed by some power of
u - μ, the maximal generalised eigenspace of u at the eigenvalue μ = a⁻¹.
Uniqueness of the Riesz projection. Two Riesz projections of u at a of the same order
are equal.
Compression of Riesz projections along a retract. Let r ∘ i = id present M as a
retract of a normed module N. If E is a Riesz projection of the extension i ∘ u ∘ r at a
of order h, then r ∘ E ∘ i is a Riesz projection of u at a of order h.
The Riesz projector 1 - (C b) ^ h of an endomorphism u of C₀(I, R) at the scalar
a, of order h, where C = 1 - a • u and b = c⁻¹ • ΔʰV(a) is the value at a of the h-th
Hasse derivative of the Fredholm resolvent, normalised by c = ΔʰH_u(a). When u is row compact
and a is a good zero of order h of the Fredholm determinant H_u, it is the unique Riesz
projection of u at a (ContinuousLinearMap.IsRowCompact.isRieszProjection_rieszProjection);
otherwise it is a junk value.
Instances For
The Riesz projector of order 0 is zero.
A Riesz projection of a row-compact operator is row compact, including when its nilpotence bound is zero.
The Riesz projector at a good zero. Let u be row compact, and let a be a good zero of
order h of its Fredholm determinant H_u: ΔʲH_u(a) = 0 for j < h and ΔʰH_u(a) is a unit.
Then 1 - (C b) ^ h is a Riesz projection of u at a of order h.
The Riesz projector is row compact. Let u be row compact and suppose ΔʰH_u(a) is a
unit. Then the Riesz projector 1 - (C b) ^ h of u at a is row compact: for h > 0 it is
(1 - C b) * ∑ i < h, (C b) ^ i, where 1 - C b = -ΔʰH_u(a)⁻¹ • u Δʰ⁻¹V(a).
A Riesz projection of a row-compact u at a good zero a of order h of its Fredholm
determinant is the explicit Riesz projector 1 - (C b) ^ h.