The Fredholm resolvent of a row-compact operator #
Let R be a complete ultrametric normed commutative ring with ‖1‖ = 1 and Tate scalars, I a
discrete index type, and u an endomorphism of the model space C₀(I, R) with Fredholm
determinant H_u(T) = det (1 - T u) = ∑ cₙ Tⁿ, the principal-minor series
u.matrixCoeff.charPowerSeries. The Fredholm resolvent of u is the power series
V(T) = ∑ Vₘ Tᵐ with operator coefficients
Vₘ = ∑ k ∈ range (m + 1), cₖ • u ^ (m - k) (ContinuousLinearMap.resolventCoeff),
the series ContinuousLinearMap.resolventSeries. Formally V(T) = H_u(T) (1 - T u)⁻¹: it
satisfies
(1 - T u) V(T) = V(T) (1 - T u) = H_u(T) • 1
(ContinuousLinearMap.one_sub_C_mul_X_mul_resolventSeries,
ContinuousLinearMap.resolventSeries_mul_one_sub_C_mul_X). For a finite matrix A it is the
adjugate of 1 - T A (Matrix.coeff_adjugate_one_sub_X_smul_apply). The definition and these
identities are algebraic and hold over any seminormed commutative ring.
The analytic content is that V is entire in operator norm when u is row compact, even
though the Neumann series (1 - T u)⁻¹ = ∑ uⁿ Tⁿ need not converge. If r i bounds the i-th
row of the matrix of u, the coefficients satisfy the same estimate as the determinant:
‖Vₘ‖ ≤ c whenever c ≥ 0 bounds ∏ i ∈ s, r i for every m-element set s
(ContinuousLinearMap.IsRowCompact.ringOpNorm_resolventCoeff_le). Consequently
‖Vₘ‖ * ρ ^ m → 0 for every ρ: the resolvent is a restricted power series at every radius,
that is, entire (ContinuousLinearMap.IsRowCompact.isEntire_resolventSeries), as is the
determinant itself (ContinuousLinearMap.IsRowCompact.isEntire_charPowerSeries).
Both series can therefore be evaluated at every scalar a. The value
V(a) = ∑ₘ aᵐ • Vₘ (ContinuousLinearMap.resolventAt, the evaluation
PowerSeries.evalScalar of V at a) converges in operator norm
(ContinuousLinearMap.IsRowCompact.hasSum_resolventAt), commutes with u
(ContinuousLinearMap.commute_resolventAt), is bounded by any uniform bound on
‖a‖ ^ m * ‖Vₘ‖ (ContinuousLinearMap.ringOpNorm_resolventAt_le), and satisfies the evaluated
identity
(1 - a • u) V(a) = V(a) (1 - a • u) = H_u(a) • 1
(ContinuousLinearMap.IsRowCompact.one_sub_smul_mul_resolventAt,
ContinuousLinearMap.IsRowCompact.resolventAt_mul_one_sub_smul), where H_u(a) is the value
PowerSeries.evalAt of the determinant at a. Hence 1 - a • u is invertible whenever H_u(a)
is a unit of R, with inverse H_u(a)⁻¹ • V(a)
(ContinuousLinearMap.IsRowCompact.isUnit_one_sub_smul,
ContinuousLinearMap.IsRowCompact.inverse_one_sub_smul_eq). This needs no inverse of any
operator in advance: the formal Neumann series ∑ aⁿ uⁿ need not converge. The value V(a) is
an operator-norm limit of polynomials in u
(ContinuousLinearMap.resolventAt_mem_closure_adjoin).
The same holds for the Hasse derivatives ΔᵏV of the resolvent (PowerSeries.hasseDeriv, which
divides by no factorials): their values ΔᵏV(a) (ContinuousLinearMap.resolventHasseDerivAt)
converge in operator norm (ContinuousLinearMap.IsRowCompact.hasSum_resolventHasseDerivAt),
commute with u and lie in the closure of the polynomials in u. Differentiating the resolvent
identity gives
(1 - a • u) Δᵏ⁺¹V(a) = Δᵏ⁺¹V(a) (1 - a • u) = Δᵏ⁺¹H_u(a) • 1 + u ΔᵏV(a)
(ContinuousLinearMap.IsRowCompact.one_sub_smul_mul_resolventHasseDerivAt_succ,
ContinuousLinearMap.IsRowCompact.resolventHasseDerivAt_succ_mul_one_sub_smul). Put
C = 1 - a • u. If ΔʲH_u(a) = 0 for j < h, it follows that
C ^ (h + 1) ΔʰV(a) = ΔʰH_u(a) • C ^ h
(ContinuousLinearMap.IsRowCompact.one_sub_smul_pow_succ_mul_resolventHasseDerivAt). If moreover
ΔʰH_u(a) is a unit, so that a is a good zero of order h of H_u, then
b = ΔʰH_u(a)⁻¹ • ΔʰV(a) satisfies C ^ h (C b - 1) = 0
(ContinuousLinearMap.IsRowCompact.one_sub_smul_pow_succ_mul_inverse_smul_resolventHasseDerivAt).
This is the identity behind the Riesz decomposition of u at a good zero: 1 - (C b) ^ h is
the Riesz projector onto the generalized eigenspace ker C ^ h, and (C b) ^ h is its
complementary projector.
The coefficient estimate ‖Vₘ‖ ≤ c is first proved for the output truncations π_S ∘ u to
finite sets S of coordinates. The entries of their resolvent coefficients are entries of the
coefficients of adjugate (1 - T A) for finite blocks A, which are coefficients of
determinants of matrices of linear polynomials
(Matrix.norm_sum_coeff_charpolyRev_smul_pow_apply_le). It then passes to u because the
truncations converge to u in operator norm, and each coefficient of the determinant, hence
each Vₘ, is continuous in operator norm on row-compact operators.
References #
- J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. IHÉS 12 (1962), 69–85, §§4–7.
- R. Coleman, p-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479, Appendix A.
The m-th coefficient ∑ k ∈ range (m + 1), cₖ • u ^ (m - k) of the Fredholm resolvent
det (1 - T u) (1 - T u)⁻¹ of an endomorphism u of C₀(I, R), where cₖ is the k-th
coefficient of the Fredholm determinant u.matrixCoeff.charPowerSeries.
Instances For
The recursion V_{m+1} = c_{m+1} • 1 + u * V_m for the coefficients of the Fredholm
resolvent.
The recursion V_{m+1} = c_{m+1} • 1 + V_m * u for the coefficients of the Fredholm
resolvent.
Each coefficient of the Fredholm resolvent commutes with the operator.
The Fredholm resolvent V(T) = det (1 - T u) (1 - T u)⁻¹ of an endomorphism u of
C₀(I, R), the power series with coefficients ContinuousLinearMap.resolventCoeff.
Instances For
The resolvent identity (1 - T u) V(T) = det (1 - T u) • 1.
The resolvent identity V(T) (1 - T u) = det (1 - T u) • 1.
The coefficients of the Hasse derivatives of the Fredholm resolvent:
coeff m (Δᵏ V) = (m + k).choose k • V_{m+k}.
The Fredholm determinant is entire. For a row-compact u, the principal-minor series
det (1 - T u) is a restricted power series at every radius.
Operator-norm bound for the Fredholm resolvent. If u is row compact, r i bounds the
i-th row of its matrix, and c ≥ 0 bounds ∏ i ∈ s, r i for every m-element set s, then
c bounds the operator norm of the m-th coefficient of the Fredholm resolvent of u. This is
the bound satisfied by the m-th coefficient of the Fredholm determinant
(Matrix.norm_charCoeff_le).
Uniform tail bound for the Fredholm resolvent. Suppose u is row compact and r i
bounds the i-th row of its matrix, and let ρ > 0, 0 ≤ q ≤ 1, 1 ≤ B, and T a finite
set of indices with ρ * r i ≤ q off T and ρ * r i ≤ B everywhere. Then
‖u.resolventCoeff m‖ * ρ ^ m ≤ B ^ #T * q ^ (m - #T), with truncated subtraction in the
exponent. The bound depends on u only through r.
The Fredholm resolvent is entire. For a row-compact u, the resolvent
det (1 - T u) (1 - T u)⁻¹ is a restricted power series at every radius in the Banach ring of
operators: ‖u.resolventCoeff m‖ * c ^ m → 0 for every c.
The value V(a) = ∑ₘ aᵐ • Vₘ at a scalar a of the Fredholm resolvent
V(T) = det (1 - T u) (1 - T u)⁻¹ of an endomorphism u of C₀(I, R), the sum being taken in
the operator norm. It takes the junk value 0 when the sum does not converge; for row-compact
u over a complete ring it converges at every a
(ContinuousLinearMap.IsRowCompact.hasSum_resolventAt).
The coefficients of the resolvent are operators rather than scalars, so this is the evaluation
PowerSeries.evalScalar at a scalar rather than PowerSeries.evalAt.
Instances For
The value of the Fredholm resolvent at a is the evaluation of the series
ContinuousLinearMap.resolventSeries at a.
At the scalar 0 the resolvent is the identity, its constant coefficient.
The value ΔᵏV(a) = ∑ₘ aᵐ • (m + k).choose k • V_{m+k} at a scalar a of the k-th Hasse
derivative of the Fredholm resolvent V(T) of an endomorphism u of C₀(I, R), the sum being
taken in the operator norm. It takes the junk value 0 when the sum does not converge; for
row-compact u over a complete ultrametric ring it converges at every a
(ContinuousLinearMap.IsRowCompact.hasSum_resolventHasseDerivAt). For k = 0 it is the value
ContinuousLinearMap.resolventAt of the resolvent itself.
Instances For
The value at a of the k-th Hasse derivative of the Fredholm resolvent is the evaluation
at a of the series PowerSeries.hasseDeriv k u.resolventSeries.
The 0-th Hasse derivative of the resolvent at a is the value V(a) of the resolvent.
At the scalar 0, the k-th Hasse derivative of the resolvent is its k-th coefficient.
The Hasse derivatives of the Fredholm resolvent at a scalar commute with the operator.
The normalised Hasse derivative b = c⁻¹ • ΔʰV(a) of the resolvent commutes with
1 - a • u.
The Hasse derivatives of the Fredholm resolvent at a scalar lie in the operator-norm closure
of the polynomials in u.
The value of the Fredholm resolvent at a scalar commutes with the operator.
The value of the Fredholm resolvent at a scalar lies in the operator-norm closure of the
polynomials in u.
Operator-norm bound for the value of the Fredholm resolvent. If c bounds
‖a‖ ^ m * ‖Vₘ‖ for every m, then it bounds the operator norm of V(a).
Hasse derivatives of the Fredholm resolvent at a scalar. For a row-compact u and every
scalar a, the series ∑ₘ aᵐ • (m + k).choose k • V_{m+k} converges in operator norm to
ΔᵏV(a).
Evaluation of the Fredholm resolvent. For a row-compact u and every scalar a, the
series ∑ₘ aᵐ • Vₘ converges in operator norm to V(a).
The evaluated resolvent identity (1 - a • u) V(a) = H_u(a) • 1, where
H_u(a) = det (1 - a u) is the value at a of the entire Fredholm determinant of the
row-compact operator u.
The evaluated resolvent identity V(a) (1 - a • u) = H_u(a) • 1 for a row-compact u.
Invertibility from the Fredholm determinant. If u is row compact and H_u(a), the
value at a of its Fredholm determinant, is a unit of R, then 1 - a • u is invertible.
The inverse of 1 - a • u. If u is row compact and H_u(a), the value at a of its
Fredholm determinant, is a unit of R, then the inverse of 1 - a • u is H_u(a)⁻¹ • V(a).
The differentiated resolvent identity (1 - a • u) Δᵏ⁺¹V(a) = Δᵏ⁺¹H_u(a) • 1 + u ΔᵏV(a)
for a row-compact u, where Δᵏ⁺¹H_u(a) is the value at a of the (k + 1)-th Hasse
derivative of the Fredholm determinant. It is the value at a of the Hasse derivative of
(1 - T u) V(T) = H_u(T) • 1.
The differentiated resolvent identity Δᵏ⁺¹V(a) (1 - a • u) = Δᵏ⁺¹H_u(a) • 1 + u ΔᵏV(a)
for a row-compact u.
The resolvent at a zero of the Fredholm determinant. Let u be row compact, put
C = 1 - a • u, and suppose the Hasse derivatives ΔʲH_u(a) vanish for j < h. Then
C ^ (h + 1) ΔʰV(a) = ΔʰH_u(a) • C ^ h. For h = 0 this is the evaluated resolvent identity.
In particular C ^ (j + 1) ΔʲV(a) = 0 for j < h.
The Riesz identity at a good zero. Let u be row compact, put C = 1 - a • u, and
suppose ΔʲH_u(a) = 0 for j < h while c = ΔʰH_u(a) is a unit, so that a is a good zero
of order h of the Fredholm determinant. Then b = c⁻¹ • ΔʰV(a) satisfies
C ^ (h + 1) b = C ^ h, that is, C ^ h (C b - 1) = 0. This is the identity from which the
Riesz projector 1 - (C b) ^ h at a and its complementary projector (C b) ^ h are built.
The normalised Hasse derivative of the resolvent inverts 1 - a • u up to a multiple of u:
(1 - a • u) (c⁻¹ • Δᵏ⁺¹V(a)) = 1 + c⁻¹ • u ΔᵏV(a) when c = Δᵏ⁺¹H_u(a) is a unit.