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Closed range and boundedness of the Hilbert pseudoinverse

Statement

Assume the Axiom of Countable Choice. Let $K : X \to Y$ be the compact linear operator of a real or complex Hilbert-space inverse model, and let $K^{†} : D (K^{†}) \to X$ be its Hilbert-space Moore–Penrose inverse, with $D (K^{†}) = R (K) \oplus R (K)^{⊥}$ carrying the norm inherited from $Y$ . Then $K^{†}$ is bounded on its domain if and only if $R (K)$ is closed in $Y$ ; for compact $K$ , these conditions are also equivalent to $R (K)$ being finite-dimensional. Here bounded means that some real $C \geq 0$ satisfies $∥ K^{†} y ∥_{X} \leq C ∥ y ∥_{Y}$ for every $y \in D (K^{†})$ .

Statement references

Hilbert-space Moore–Penrose inverse and its domain

Clason, Theorem 3.7 and Corollary 3.8, §3.1, printed pp. 20–21

Scherzer and Mindrinos, Theorem 6.6, printed p. 86 (singular-system context)

Facts & Assumptions

Given: Countable Choice, a compact linear map $K : X \to Y$ between real or complex Hilbert spaces, and its Moore–Penrose inverse $K^{†}$ .

[F1]

The compact model has Hilbert parameter/data spaces and a compact linear forward map $K$ (Compact linear inverse model between Hilbert spaces).

[F2]

$D (K^{†}) = R (K) \oplus R (K)^{⊥}$ , and on $r + z$ the inverse is the minimum-norm preimage of $r$ (Hilbert-space Moore–Penrose inverse).

[F3]

Under Countable Choice, the compact singular system has a finite positive index set exactly in the finite-dimensional-range case; otherwise its positive singular values are indexed by $J = {1, 2, 3, \dots}$ and form a sequence tending to zero, with orthonormal singular vectors and $K v_{j} = s_{j} u_{j}$ , $K^{*} u_{j} = s_{j} v_{j}$ (Singular value decomposition for compact operators).

[F4]

The adjoint identity is $⟨ K x, u ⟩_{Y} = ⟨ x, K^{*} u ⟩_{X}$ (The Hilbert-space adjoint of a bounded operator).

[F5]

For a finite orthonormal family $(e_{i})$ , $\sum_{i} ∣ ⟨ x, e_{i} ⟩ ∣^{2} \leq ∥ x ∥^{2}$ (The finite Bessel inequality and best approximation by a finite orthonormal family).

[F6]

Finite orthogonal sums satisfy Pythagoras (Pythagoras and finite orthogonal sums).

[F7]

A finite-dimensional subspace of a normed space is closed (A finite-dimensional normed subspace is closed).

[F8]

A Hilbert space is complete in its induced norm (Hilbert space).

[F9]

Cauchy–Schwarz makes $y \mapsto ⟨ y, u ⟩$ continuous for fixed $u$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F10]

For every real $ε > 0$ there is a natural $n \geq 1$ with $1 / n < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[F11]

A starting value and a total recursive rule define a unique sequence (The recursion theorem).

[F12]

Every nonempty subset of $N$ has a least element (The well-ordering principle).

[A1]

Countable Choice is the hypothesis of the compact SVD in [F3] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF1F2F3F5F6F7algebra

Suppose the positive singular system is finite, with index set $J = {1, \dots, r}$ . If $r = 0$ , the SVD expansion gives $K = 0$ , so $R (K) = {0}$ is closed by [F7] and $K^{†} = 0$ is bounded on its domain. If $r > 0$ , each $u_{j} = s_{j}^{- 1} K v_{j}$ lies in $R (K)$ , while the SVD expansion places every $K x$ in $span {u_{1}, \dots, u_{r}}$ ; hence the range equals this finite-dimensional span and is closed by [F7]. For $y = q + z \in D (K^{†})$ , $q \in R (K)$ and $z \in R (K)^{⊥}$ , so $K^{†} y = \sum_{j = 1}^{r} s_{j}^{- 1} ⟨ q, u_{j} ⟩ v_{j}$ by [F2] and [F3]. Pythagoras and Bessel give $∥ K^{†} y ∥^{2} = \sum_{j = 1}^{r} s_{j}^{- 2} ∣ ⟨ q, u_{j} ⟩ ∣^{2} \leq s_{r}^{- 2} ∥ q ∥^{2} \leq s_{r}^{- 2} ∥ y ∥^{2}$ ; here $s_{r}$ is the smallest positive singular value, and the last inequality follows from $q ⊥ z$ . Thus finite rank implies closed range and bounded $K^{†}$ .

1.2givenF3F4F5F6F8F9F10F11F12A1chooseconstructalgebra

Suppose instead that the positive singular index set is $J = {1, 2, 3, \dots}$ . Since $s_{j} \to 0$ , for each pair $(k, n)$ the set of integers $m > n$ with $s_{m} < 2^{- (k + 1)}$ is nonempty; [F12] gives its least element, so [F11] defines recursively $n_{0} = 0 < n_{1} < n_{2} < \dots$ and $s_{n_{k}} < 2^{- k}$ for every $k \geq 1$ . Set $y_{N} = \sum_{k = 1}^{N} s_{n_{k}} u_{n_{k}}$ . For $M > N$ , [F6] gives $∥ y_{M} - y_{N} ∥^{2} = \sum_{k = N + 1}^{M} s_{n_{k}}^{2} \leq \sum_{k = N + 1}^{M} 4^{- k} \leq 4^{- N} / 3$ , so $(y_{N})$ is Cauchy and converges to some $y \in Y$ by [F8]. Each $y_{N}$ belongs to $R (K)$ because $s_{n_{k}} u_{n_{k}} = K v_{n_{k}}$ , hence $y \in \overline{R (K)}$ . If $y = K x$ , then for every $k \geq 1$ and every $N \geq k$ , orthonormality gives $⟨ y_{N}, u_{n_{k}} ⟩ = s_{n_{k}}$ ; by [F9] we may pass to the limit to get $⟨ y, u_{n_{k}} ⟩ = s_{n_{k}}$ . The SVD and adjoint identity [F3]–[F4] then give $s_{n_{k}} = ⟨ K x, u_{n_{k}} ⟩ = s_{n_{k}} ⟨ x, v_{n_{k}} ⟩$ , so $⟨ x, v_{n_{k}} ⟩ = 1$ for every $k \geq 1$ . Applying [F5] to any first $N$ of these orthonormal vectors gives $N \leq ∥ x ∥^{2}$ for every $N \geq 1$ . Taking $ε = 1 / (∥ x ∥^{2} + 1)$ in [F10] yields an integer $N \geq 1$ with $1 / N < 1 / (∥ x ∥^{2} + 1)$ , hence $N > ∥ x ∥^{2}$ , a contradiction. Thus $y \notin R (K)$ and the range is not closed.

2.1step 1.2F2F3algebra

For each $k \geq 1$ , $K v_{n_{k}} = s_{n_{k}} u_{n_{k}} \in D (K^{†})$ and $K^{†} K v_{n_{k}} = v_{n_{k}}$ by [F2], since $v_{n_{k}} \in N (K)^{⊥}$ . Therefore $∥ K^{†} K v_{n_{k}} ∥ / ∥ K v_{n_{k}} ∥ = 1 / s_{n_{k}} > 2^{k}$ , so $K^{†}$ is unbounded when the positive singular system is infinite.

3.1step 1.1step 1.2step 2.1F3∎

By [F3], every compact $K$ has either finite range dimension, covered by step 1.1, or an infinite positive singular system, covered by steps 1.2 and 2.1. Thus $R (K)$ is closed exactly in the finite-dimensional case, and in exactly that case $K^{†}$ is bounded. This proves both stated equivalences.

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Sources

Christian Clason, Regularization of Inverse Problems: Lecture Notes, arXiv:2001.00617v2, Theorem 3.7 (standard reference, not scraped)
Otmar Scherzer and Leonidas Mindrinos, Inverse Problems: Lecture Notes, University of Vienna, 2015, Theorem 6.6 (singular-system context) (standard reference, not scraped)