Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

For a one-parameter family of simple metrics of constant curvature ($4\kappa$ for $\kappa\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions {\it \`a la} Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted $L^2-L^2$ setting which is equivalent to the $L^2(M, dVol_\kappa)\to L^2(\partial_+ SM, d\Sigma^2)$ one for any $\kappa\in (-1,1)$.


Introduction
Our object of study is the geodesic X-ray transform on a special family of simple surfaces. To give some context, fix a Riemannian surface (M, g), with strictly convex boundary and no infinite-length geodesic. Denote its unit circle bundle SM := {(x, v) ∈ T M, g x (v, v) = 1}. The manifold of geodesics can then be modelled over the inward boundary ∂ + SM (points in SM such that x ∈ ∂M and v points inwards), carrying a natural measure dΣ 2 (the volume form coming from the Sasaki metric on SM ). In this context, one defines the geodesic X-ray transform I 0 : C ∞ (M ) → C ∞ (∂ + SM ) as where γ x,v (t) is the unit-speed geodesic with γ(0) = x andγ(0) = v, and τ (x, v) is its first exit time. In integral geometry, one is concerned with the reconstruction of f from knowledge of constant curvature 4κ for any fixed κ ∈ (−1, 1). Specifically, we establish the singular value decomposition of the operators P − and C − when viewed as operators from L 2 (∂ + SM, dΣ 2 ) into itself, see Theorem 10 below. This in particular alllows to formulate a few range characterizations of I 0 . First note that on functions on ∂ + SM , all the action takes place in the subspace of those invariant by the map S A (8) (this is because integrating a function does not depend on the direction of the geodesic).
Outline. The remainder of the article is structured as follows: in Section 2, we first introduce the geometric models considered and compute their scattering relation, involving in particular an important function s κ (α) (equal to α in the Euclidean case); in Section 3, we construct the SVD's of the operators P − and C − , which help describe the range of the geodesic X-ray transform in Theorem 1; finally, in Section 4, we construct the SVD of an appropriate adjoint of I 0 , and give a proof of Theorem 2.

Remark 4 (On notation).
In what follows, we will always work with one fixed value of κ, and all quantities are κ-dependent, whether specified in the notation or not. The following may give a sample of which ones generally include κ in the notation and which ones don't: dΣ 2 , g κ , dV ol κ , s κ , w κ , Z κ n,k , ψ κ n,k , σ κ n,k , C − , P − , SM, S, S A , A ± , A * ± , I 0 , I ♯ 0 .
For fixed κ ∈ (0, 1), the manifold (M, g κ ) can be viewed as a simple surface included in the "sphere" (C∪{∞}, g κ ) and for κ ∈ (−1, 0), the manifold (M, g κ ) can be viewed as a simple surface included in the hyperbolic space ( In either case, κ → 0 recovers the standard Euclidean disk. As |κ| → 1, simplicity breaks down for two different reasons: (M, g 1 ) becomes a "hemisphere" with totally geodesic (i.e., nonconvex) boundary and (M, g −1 ) is, up to some scalar constant 1 , the Poincaré disk, non-compact. In the latter, the interior of M is geodesically complete, all geodesics are asymptotically normal to the boundary and the fan-beam coordinate system breaks down.
To compute geodesics, we will use the action of isometries of either model, to move the following obvious geodesics One can find those isometries by conjugating the automorphisms of the Poincaré disk or the Riemann sphere with appropriate homotheties, which would result in subgroups of Möbius transformations. Under this latter assumption, let us find those directly, with the immediate observation that a Möbius transformation T (z) = az+c cz+d pushes forward a tangent vector (z, ζ) to T · (z, ζ) = (T (z), T ′ (z)ζ). We will also write T (z) = az+b cz+d = a b c d (z) interchangeably. Lemma 5. For κ ∈ (0, 1), the isometry group of (C ∪ {∞}, g κ ) is given by For κ ∈ (−1, 0), the isometry group of (D (−κ) −1/2 , g κ ) is given by Proof. The proofs of (3) and (4) are identical. We seek a Möbius transformation T = a b c d with ad − bc = 1 such that g κ (T (z))(T ′ (z)ζ, T ′ (z)ζ) = g κ (z)(ζ, ζ) for all (z, ζ). This is recast as 1 Customarily, the Poincaré disk carries four times this metric.
which yields, for all z in the space considered This is equivalent to having the relations Multiplying the second byā and using the first and ad − bc = 1, we get Similarly, multiplying the same equation byc yields Finally, these two relations are necessary and sufficient to describe (3) and (4). Now, given (z 1 , θ) corresponding to a unit tangent vector (z 1 , c κ (z 1 )e iθ ), we want to find the element T which maps (0, 1) to (z 1 , c κ (z 1 )e iθ ), satisfying Seeking for an element of the form (3) or (4) immediately leads to the unique transformation

Scattering relation
We generally define the scattering relation S : ∂SM → ∂SM as ) denotes the geodesic flow on a Riemannian manifold (M, g) and τ (x, v) denotes the first exit time of the geodesic γ x,v (t). In our case, we now compute this relation explicitly: We first compute the geodesic through the point (1, c κ (1)e i(π+α) ) with α ∈ (−π/2, π/2). To do this, we first compute that the unique isometry mapping (0, 1) to that point is given by (2) is the geodesic we seek. We then solve for |T (z(t * ))| 2 = 1 with t * > 0, the point at which that geodesic exists the domain M . By direct computation, this yields and subsequently The number inside the argument belongs to the right-half plane so that we may compute that T (z(t * )) = exp i π + 2 tan −1 1 − κ 1 + κ tan α .

Scattering signatures
The function s = s κ defined as may be thought of as a 'scattering signature' of each geometry (the only function that distinguishes two circularly symmetric scattering relations on the unit disk). Strikingly, we have s κ • s −κ = id for all κ ∈ (−1, 1). As we will work with only one fixed value of κ at a time, we may drop the subscript κ for conciseness. The scattering relation S and antipodal scattering relation S A (composition of S with the antipodal map α → α + π) take the form S(β, α) = (β + π + 2s(α), π − α) , S A (β, α) = (β + π + 2s(α), −α) .
The map S A is a diffeomorphism of ∂SM , and ∂ ± SM are both S A -stable. Since integrating a function does not depend on the direction of integration, the ray transform of a function is always invariant under the pullback S * A . Note that the function s(α) satisfies the following obvious properties: The jacobian of α → s(α) takes the expression in particular, 1 λ ≤ s ′ (α) ≤ λ for all α and s ′ (α) can be used as a multiplicative weight on L 2 spaces, that yields an equivalent L 2 topology. In the Euclidean case, s(α) = α, and therefore no distinction is necessary. In the work that follows, it will be crucial to work with α, s(α) or a combination of both. To this end, we now describe some important relations between the two.

Linear fractional relation between e 2iα and e 2is(α) and its consequences
An important calculation is the following: with The following Lemma will be crucial. Below we will say that a function f (α) is a holomorphic/strictly holomorphic/antiholomorphic/striclty antiholomorphic in e iα if its Fourier expansion in e iα only contains non-negative/positive/non-positive/negative powers of e iα . Lemma 6. For any κ ∈ (−1, 1), the function e 2is(α) is a holomorphic, even series in e iα , with average κ. As a result, for any q > 0, e 2iqs(α) is a holomorphic, even series in e iα , and for q < 0, e 2iqs(α) is an anti-holomorphic, even series in e iα .
Proof. Use a geometric sum in the expression above to obtain The other consequences follow from the fact that products of holomorphic series are holomorphic.

Relation between e iα and e is(α)
While there is no obvious relation between e iα and e is(α) (and it is unclear whether e is(α) is holomorphic in terms of e iα ), some crucial relations are to be derived. A first one is that √ s ′ can be writen as an expression of both e iα and e is(α) .

Lemma 7.
With s(α) = s κ (α) as given in (7), we have Proof. Recall the formula , then an immediate calculation shows that Further, notice that So f is in fact real-valued, and using (14), it is nothing but Multiplying (13) by e −iα and identifying real and imaginary parts, we obtain relations for the sines and cosines: 3 Singular Value Decomposition of the boundary operators and moment conditions for I 0 .
Out of the scattering relation (5), one defines operators with ν x the unit inner normal to x ∈ ∂M , in particular in fan-beam coordinates, this is nothing but cos α. In the circularly symmetric case, since µ(S(x, v)) = −µ(x, v), A ± and A * ± are also adjoints of one another in the L 2 (∂ + SM, dΣ 2 ) → L 2 (∂SM, dΣ 2 ) setting. In the smooth setting, as such extensions may generate singularities at the tangential directions, one must define, somewhat tautologically for now, see Appendix A for more detail, and for their further decompositions into spaces C ∞ α,±,± (∂ + SM ) in Eq. (50). We define the fiberwise Hilbert transform H : C ∞ (∂SM ) → C ∞ (∂SM ), defined in fan-beam coordinates as and write H = H + + H − , where H +/− is the restriction of H onto even/odd Fourier modes. Out of these operators, we can then define two important operators One of the purposes of this section will be to compute the SVD's of P − and C − for the L 2 (∂ + SM, dΣ 2 ) → L 2 (∂ + SM, dΣ 2 ) topology. The relevance of these operators comes from the range characterization described in Proposition 18, which tell us that understanding the range of I 0 reduces to understanding the range of P − on C ∞ α,+,− (∂ + SM ). Moreover, understanding C − provides another range characterization for I 0 , together with operators for projecting noisy data onto the range of I 0 .
In Section 3.1, we first give a characterization of the spaces C ∞ α,±,± (∂ + SM ) in terms of 'natural', distinguished bases. We then modify these bases in Section 3.2 so as to construct the SVD's of P − and C − . Finally in Section 3.3, we then formulate the range characterizations of I 0 , together with some consequences and applications.

Description of the spaces
In cases where the scattering relation admits an explicit expression, we can construct bases for C ∞ α,±,± (∂ + SM ) defined in Eq. (50) using appropriate Fourier series, ruling out some coefficients by symmetry arguments. Upon defining the family we can formulate the following Proof. Let u ∈ C ∞ (∂SM ). Upon expanding the smooth function u(β, s −1 (α)) in Fourier series in (β, α), we obtain an expansion with rapid decay for u, of the form Upon looking at e p,ℓ defined in (18), we find that so that At the level of the Fourier coefficients, this means For σ 1 = σ 2 , the second equality forces u p,ℓ = 0 for all ℓ odd, and using the first equality implies (19) and (22) upon writing ℓ = 2q. For σ 1 = σ 2 , the second equality forces u p,ℓ = 0 for all ℓ even, and using the first equality implies (20) and (21) upon writing ℓ = 2q + 1.

Singular value decompositions of P − and C −
Recall the definitions (17) of P − and C − , where according to Appendix A, P − is naturally defined on C ∞ α,+,− (∂ + SM ) and C − is naturally defined on C ∞ α,−,+ (∂ + SM ). Functions which transform well under P − or C − must be nicely compatible with both the fiberwise Hilbert transform (16) and the scattering relation (5). The bases displayed in (20) and (21) do the latter but not the former. These are naturally orthogonal in L 2 (∂SM, s ′ (α) dΣ 2 ), and to make them orthogonal in L 2 (∂SM, dΣ 2 ) (a space where iH − is naturally self-adjoint), a natural modification is to multiply these bases by s ′ (α). Let us then define, for p, q ∈ Z, Combining (23) with the fact that we immediately obtain for every (p, q) ∈ Z 2 , Regarding φ ′ p,q as fiberwise odd functions on ∂SM , their fiberwise Hilbert transform can be computed, using in an important way the √ s ′ factor.
is, by virtue of Lemma 6, e iα times a fiber-holomorphic series, so it is strictly holomorphic and as such satisfies . By virtue of Lemma 6 again, the last factor is antiholomorphic, while upon complex conjugating (10), is a strictly antiholomorphic series. The product is thus strictly antiholomorphic in e iα , therefore Hφ ′ p,q = iφ ′ p,q . The formula follows.
Constructing functions with symmetries under S * A , we then define Upon removing these redundancies in the set of indices, we can rewrite (20) and (21) as Finally, we note how the basis elements φ ′ p,q transform under id − S * : Now, given the properties satisfied by φ ′ p,q , u ′ p,q , v ′ p,q , the action of H − and S * and S * A on them are formally identical as in the Euclidean case, and the same calculation as in [21, p. 444] allows to deduce that for any (p, q) in the appropriate range, Since the families {u ′ p,q } and {v ′ p,q } are orthogonal in L 2 (∂ + SM, dΣ 2 ), this automatically produces the singular value decompositions of P − and C − , viewed as operators from that space into itself. The statements are identical to those of the Euclidean case made in [21, Prop. 1 and 2] (except that the definitions of u ′ p,q and v ′ p,q differ from [21] by a fixed constant). Below we denote the orthogonal splitting Theorem 10. Given κ ∈ (−1, 1), let M be the unit disk equipped with the metric g κ (1) and define P − , C − as in (17). The SVD of the operator P − : V − → V + is given by: for any (p, q) ∈ Z 2 with p < 2q + 1, The eigendecomposition of C − : V + → V + is given by: for any (p, q) ∈ Z 2 with p < 2q, Projection of noisy data onto the range of I 0 . In addition, for purposes of projection of noisy data onto the range of I 0 , an immediate consequence of Theorem 10 is the following :

Consequences of
Theorem 11. Let M be equipped with the metric g κ for κ ∈ (−1, 1) fixed, and define C − as in (17). Then the operator id + C 2 − is the L 2 (∂ + SM, dΣ 2 ) orthogonal projection operator onto the range of I 0 .

Singular Value Decomposition of the X-ray transform
A conclusion of Theorem 1 is that the range of I 0 is spanned by an orthogonal family in V + . In what follows, the goal is to apply an appropriate adjoint for I 0 to the family (26), and find a topology for which the functions obtained are orthogonal. Most adjoints for I 0 are constructed out of a distinguished one which we denote I ♯ 0 : it corresponds to the adjoint of I 0 : L 2 (M, dV ol κ ) → L 2 (∂ + SM, µ dΣ 2 ), which in our setting takes the expression where (β − , α − )(z, θ) are the unique fan-beam coordinates of the unique g κ -geodesic passing through (z, θ) ∈ SM , or 'footpoint map'.
In what follows, we will first recall in Section 4.1 what is known in the Euclidean case, before showing that combining this knowledge with our previous derivations ultimately allows to produce the SVD of the X-ray transform in Section 4.2. Proofs of some intermediary lemmas are relegated to Section 4.3.

Euclidean case -Zernike polynomials
It may be convenient to reparameterize the set (26) to make the Zernike basis appear, in the form that it is presented in [9]. Specifically, for n ∈ N and k ∈ Z, we reparameterize the basis of V + as ψ n,k := (−1) n 4π u ′ n−2k,n−k instead, i.e. we have involved the change of index (n, k) → (p, q) = (n − 2k, n − k), n ∈ N 0 , k ∈ Z.
Then an immediate calculation yields and we now want to compute I ♯ 0 ψ n,k µ . Together with the definition of I ♯ 0 and the relations satisfied by the Euclidean footpoint map for all (ρe iω , θ) ∈ SM : we arrive at the expression With the relation sin α − (ρ, θ) = −ρ sin θ, we may rewrite this as where we have defined The functions W n are related to the Chebychev polynomials of the second kind U n , specifically through the relation W n (t) = i n U n (t). In particular, it is immediate to check the 2-step recursion relation and initial conditions By induction, the top-degree term of W n is (2it) n . Fixing n ≥ 0, we now split the calculation into two cases: Case k < 0 or k > n. In light of (29), since W n is a polynomial of degree n, then W n (−ρ sin θ) is a trigonometric polynomial of degree n in e iθ . In particular, if k < 0 or k > n, then |n−2k| > n and thus the right hand side of (29) is identically zero. In short, we deduce Case 0 ≤ k ≤ n. For the remaining cases, we then define Z n,k := I ♯ 0 ψ n,k cos α , and for the sake of self-containment, we now show that the functions {Z n,k } n≥0, 0≤k≤n so constructed are the Zernike basis in the convention of [9], by showing that they satisfy Cauchy-Riemann systems and take the same boundary values.
Lemma 12. The functions {Z n,k } n≥0, 0≤k≤n satisfy the following properties: For all n ≥ 0 Proof. Using the relation W n (−t) = (−1) n W n (t), we arrive at the expression With ∂ z = e −iω 2 (∂ ρ − i ρ ∂ ω ) and ∂z = e iω 2 (∂ ρ + i ρ ∂ ω ), we compute Plugging these into (33) immediately implies In addition, we compute where the second equality comes from the fact that the lower-order terms of W n (ρ sin θ) have no harmonic content along e inθ . Finally, the constant is In short, Z n,0 = ρ n e inω = z n . This also implies ∂ z Z n,0 = 0 and since we have Z n,n = (−1) n Z n,0 = (−1) n z n , we deduce that ∂ z Z n,n = 0.
To prove the boundary condition, using that Z n,k (ρe iω ) = e i(n−2k)ω Z n,k (ρ), it is enough to show that Z n,k (1) = (−1) k for every n ≥ 0 and 0 ≤ k ≤ n. That this is true for k = 0 and k = n follows from the expressions just computed, and the general claim follows by induction on n once the following equality is satisfied: To prove (35), it suffices to input the recursion W n (sin θ) = 2i sin θW n−1 (sin θ) + W n−2 (sin θ) into the expression (33) evaluated at ρe iω = 1.
From Lemma 12, we see that the family so defined satisfies the characterization (b) of [9, Theorem 1] of the Zernike polynomials. One may see that this characterization defines the same family due the following facts: for n ≥ 0 and k = 0, the functions Z n,k in both sets agree; by induction on k > 0, in both sets of functions, Z n,k satisfies a ∂ z equation with same right-hand side and same boundary condition, for which a solution is unique if it exists.
We can then use some of the properties given in [9], in particular, the following orthogonality property and the fact that √ n+1 √ π Z n,k n≥0, 0≤k≤n is an orthonormal basis of L 2 (M ).
For the topology L 2 (∂ + SM, dΣ 2 ), the adjoint of I 0 is given by w → I ♯ 0 w µ with I ♯ 0 defined in (27). Let us then consider the functions where (β − , α − ) are short for (β − (ρe iω , θ), α − (ρe iω , θ)), the fan-beam coordinates of the unique g κ -geodesic passing through (ρe iω , θ). With the identities (15), this can be rewritten as Using the symmetries we obtain the expression with W n defined in (30), and where (α − , β − ) are evaluated at (ρ, θ). We now need to make the functions β − + s(α − ) and sin(s(α − )) more explicit. Specifically, we will derive the following in the next section: Lemma 14. The following relations hold: In light of (39), we want to make in (38) the change of variable in the fiber We then state two important identities, also proved in the next section: Lemma 15. The change of variable θ → θ ′ in (41) satisfies the following: Combining (43) with (40), we arrive at the relation Using these relations with (38), we then arrive at We now split cases in a similar way as the Euclidean case.
Case k < 0 or k > n. In light of (44), since W n is a polynomial of degree n, then the function W n − 1−κ 1−κρ 2 ρ sin θ ′ is a trigonometric polynomial of degree n in e iθ ′ . In particular, if k < 0 or k > n, then |n − 2k| > n and thus the right hand side of (44) is identically zero, and we conclude that Case 0 ≤ k ≤ n. When 0 ≤ k ≤ n, we then define Z κ n,k := I ♯ 0 ψ κ n,k µ and comparing (44) with (29), we find that in other words, for any n ≥ 0 and 0 ≤ k ≤ n, Orthogonality of Z κ n,k . Now that we fully understand the action of I ♯ 0 1 µ on V + , the last question is then to find out for which topology on M the family {Z κ n,k } is orthogonal. We look for a measure of the form w(ρ) ρ dρ dω (1+κρ 2 ) 2 , and want to change variable ρ ′ = 1−κ 1−κρ 2 ρ, with jacobian In light of the jacobian, the change ρ → ρ ′ will land in the Euclidean volume form if w(ρ) = 1+κρ 2 1−κρ 2 . Assuming this is the case, we obtain, upon using (36), Now Theorem 16 below and the proof of Theorem 2 will be based on the following observation: let (H 1 , · 1 ), (H 2 , · 2 ) be two Hilbert spaces and A : H 1 → H 2 be a bounded operator; if there exist two complete orthogonal systems {x n } in H 1 and {y n } in H 2 such that Ax n = y n for all n, then the singular value decomposition of A is (x n / x n 1 , y n / y n 2 , y n 2 / x n 1 ) n . This also implies that the SVD of the adjoint A * is (y n / y n 2 , x n / x n 1 , y n 2 / x n 1 ) n .
Based on this observation and the earlier calculations, we can formulate the following result: Theorem 16. Let κ ∈ (−1, 1). Define the weight w κ (z) := 1+κ|z| 2 1−κ|z| 2 for z ∈ M . Then the operator has kernel and its restriction to the orthocomplement of that kernel has SVD ( ψ κ n,k , Z κ n,k , σ κ n,k ) n≥0, 0≤k≤n , where and where the spectral values equal The proof of Theorem 2 now becomes straightforward.
Proof of Theorem 2. In light of Theorem 16, the SVD of the adjoint of I ♯ 0 1 µ just consists of interchanging the families ψ κ n,k , Z κ n,k , and this is the operator we are interested in. We now compute In other words, the adjoint of the operator I ♯ 0 1 µ : L 2 (∂ + SM, dΣ 2 ) → L 2 (M, w κ dV ol κ ) is the operator A : L 2 (M, w κ dV ol κ ) → L 2 (∂ + SM, dΣ 2 ), Af := I 0 (w κ f ).
In particular, the relation A Z κ n,k = σ κ n,k ψ κ n,k implies I 0 w κ Z κ n,k = σ κ n,k ψ κ n,k for all n, k. Now, given f ∈ w κ L 2 (M, w κ dV ol κ ), f wκ expands into the basis Z κ n,k , f w κ = n≥0 n k=0 a n,k Z κ n,k , where a n,k = Then we compute directly n,k a n,k AZ κ n,k = n,k a n,k σ κ n,k ψ κ n,k .
hence the result.
Together with the relation cos(2θ) = −2 1 0 1 (sin 2 θ), this implies the relation Together with the fact that sin θ and sin θ ′ have simultaneously the same sign, (43) follows upon taking squareroots.
A Spaces C ∞ α,±,± (∂ + SM), operators P ± , C ± and a refinement of the Pestov-Uhlmann range characterization In this section, we work on a general simple surface (M, g) with inward boundary ∂ + SM . The objects of study are the geodesic X-ray transforms I 0 : C ∞ (M ) → C ∞ (∂ + SM ) and I 1 : C ∞ (M ; T M ) → C ∞ (∂ + SM ), defined for any (x, v) ∈ ∂ + SM as where f is a smooth function, h is a smooth vector field, (γ x,v (t),γ x,v (t)) is the unit speed geodesic with (γ x,v (0),γ x,v (0)) = (x, v), and τ (x, v) is its first exit time.
The Pestov-Uhlmann range characterization of I 0 and I 1 appearing in [24,Theorem 4.4] relates the ranges of I 0 and I 1 with those of P − and P + as defined on We would like to restrict C ∞ α (∂ + SM ) to a 'half'-subspace incorporating a natural symmetry associated to whether one is integrating a function or a one-form. Namely, a function u in the range of I 0 satisfies S * A u = u and a function u in the range of I 1 satisfies S * A u = −u. One must also encode whether extension from ∂ + SM to ∂SM through A ± produces smooth functions.
Lemma 17. The spaces C ∞ α,± (∂ + SM ) are stable under the pull-back S * A . Proof. The map S A is the composition of the scattering relation S and the antipodal map (x, v) → (x, −v), as such it can be regarded as a smooth diffeomorphism of ∂SM , thus S * A can be viewed as an operator on C ∞ (∂ + SM ) or on C ∞ (∂SM ). Moreover, we have the relations S * A A ± = A ± S * A . In particular, if w ∈ C ∞ α,± (∂ + SM ), then A ± w is smooth on ∂SM . Then so is S * A A ± w = A ± (S * A w), which exactly means that S * A w ∈ C ∞ α,± (∂ + SM ).
Recall then the definitions of the boundary operators The spaces above provide natural smooth functional settings for these operators: • the operators P ± are naturally defined on C ∞ α,+ (∂ + SM ) and in the direct decomposition w = w + + w − , (where S * A w ± = ±w ± ), we get: P + w = P + w + ∈ ker(id + S * A ) (P + w − = 0), P − w = P − w − ∈ ker(id − S * A ) (P − w + = 0).