On Krylov solutions to infinite-dimensional inverse linear problems

Abstract

We discuss, in the context of inverse linear problems in Hilbert space, the notion of the associated infinite-dimensional Krylov subspace and we produce necessary and sufficient conditions for the Krylov-solvability of a given inverse problem, together with a series of model examples and numerical experiments.

Appendix A. Some prototypical example operators

Let us review in this Appendix certain operators in Hilbert space that were useful in the course of our discussion, both as a source of examples or counter-examples, and as a playground to understand certain mechanisms typical of the infinite dimensionality.

1.1 A.1. The multiplication operator on $\ell ^2({\mathbb {N}})$

Let us denote with $(e_n)_{n\in \mathbb {N}}$ the canonical orthonormal basis of $\ell ^2({\mathbb {N}})$. For a given bounded sequence $a\equiv (a_n)_{n\in \mathbb {N}}$ in ${\mathbb {C}}$, the multiplication by a is the operator $M^{(a)}:\ell ^2({\mathbb {N}})\rightarrow \ell ^2({\mathbb {N}})$ defined by $M^{(a)}e_n=a_n e_n$ $\forall n\in {\mathbb {N}}$ and then extended by linearity and density, in other words the operator given by the series

$$\begin{aligned} M^{(a)}\;=\;\sum _{n=1}^\infty a_n|e_{n}\rangle \langle e_n| \end{aligned}$$

(A.1)

(that converges strongly in the operator sense).

$M^{(a)}$ is bounded with norm $\Vert M^{(a)}\Vert _{\mathrm {op}}=\sup _n|a_n|$ and spectrum $\sigma (M^{(a)})$ given by the closure in ${\mathbb {C}}$ of the set $\{a_1,a_2,a_3\dots \}$. Its adjoint is the multiplication by $a^*$. Thus, $M^{(a)}$ is normal. $M^{(a)}$ is self-adjoint whenever a is real and it is compact if $\lim _{n\rightarrow \infty }a_n=0$.

1.2 A.2. The right-shift operator on $\ell ^2({\mathbb {N}})$

The operator $R:\ell ^2({\mathbb {N}})\rightarrow \ell ^2({\mathbb {N}})$ defined by $Re_n=e_{n+1}$ $\forall n\in {\mathbb {N}}$ and then extended by linearity and density, in other words the operator given by the series

$$\begin{aligned} R\;=\;\sum _{n=1}^\infty |e_{n+1}\rangle \langle e_n| \end{aligned}$$

(A.2)

(that converges strongly in the operator sense), is called the right-shift operator.

R is an isometry (i.e., it is norm-preserving) with closed range $\mathrm {ran} R=\{e_1\}^\perp$. In particular, it is bounded with $\Vert R\Vert _{\mathrm {op}}=1$, yet not compact, it is injective, and invertible on its range, with bounded inverse

$$\begin{aligned} R^{-1}:\mathrm {ran} R\rightarrow {\mathcal {H}},\qquad R^{-1}\;=\;\sum _{n=1}^\infty |e_n\rangle \langle e_{n+1}|. \end{aligned}$$

(A.3)

The adjoint of R on ${\mathcal {H}}$ is the so-called left-shift operator, namely the everywhere defined and bounded operator $L:{\mathcal {H}}\rightarrow {\mathcal {H}}$ defined by the (strongly convergent, in the operator sense) series

$$\begin{aligned} L\;=\;\sum _{n=1}^\infty |e_n\rangle \langle e_{n+1}|,\qquad L=R^*. \end{aligned}$$

(A.4)

Thus, L inverts R on $\mathrm {ran} R$, i.e., $LR=\mathbb {1}$, yet $RL=\mathbb {1}-|e_1\rangle \langle e_1|$. One has $\ker R^*=\mathrm {span}\{e_1\}$.

R and L have the same spectrum $\sigma (R)=\sigma (L)=\{z\in {\mathbb {C}}\,|\,|z|\leqslant 1\}$, but R has no eigenvalue, whereas the eigenvalue of L form the open unit ball $\{z\in {\mathbb {C}}\,|\,|z|< 1\}$.

1.3 A.3. The compact (weighted) right-shift operator on $\ell ^2({\mathbb {N}})$

This is the operator ${\mathcal {R}}:\ell ^2({\mathbb {N}})\rightarrow \ell ^2({\mathbb {N}})$ defined by the operator-norm convergent series

$$\begin{aligned} {\mathcal {R}}\;=\;\sum _{n=1}^\infty \sigma _n|e_{n+1}\rangle \langle e_n|, \end{aligned}$$

(A.5)

where $\sigma \equiv (\sigma _n)_{n\in \mathbb {N}}$ is a given bounded sequence with $0<\sigma _{n+1}<\sigma _n$ $\forall n\in {\mathbb {N}}$ and $\lim _{n\rightarrow \infty }\sigma _n=0$. Thus, ${\mathcal {R}}e_n=\sigma _n e_{n+1}$.

${\mathcal {R}}$ is injective and compact, and (A.5) is its singular value decomposition, with norm $\Vert {\mathcal {R}}\Vert _{\mathrm {op}}=\sigma _1$, $\overline{\mathrm {ran}\,{\mathcal {R}}}=\{e_1\}^\perp$, and adjoint

$$\begin{aligned} {\mathcal {R}}^*\;=\;{\mathcal {L}}\;=\;\sum _{n=1}^\infty \sigma _n|e_n\rangle \langle e_{n+1}|. \end{aligned}$$

(A.6)

Thus, ${\mathcal {L}}{\mathcal {R}}=M^{(\sigma ^2)}$, the operator of multiplication by $(\sigma _n^2)_{n\in {\mathbb {N}}}$, whereas ${\mathcal {R}}{\mathcal {L}}=M^{(\sigma ^2)}-\sigma _1^2|e_1\rangle \langle e_1|$.

1.4 A.4. The compact (weighted) right-shift operator on $\ell ^2({\mathbb {Z}})$

This is the operator ${\mathcal {R}}:\ell ^2({\mathbb {Z}})\rightarrow \ell ^2({\mathbb {Z}})$ defined by the operator-norm convergent series

$$\begin{aligned} {\mathcal {R}}\;=\;\sum _{n\in {\mathbb {Z}}}\sigma _{|n|}\,|e_{n+1}\rangle \langle e_n|, \end{aligned}$$

(A.7)

where $\sigma \equiv (\sigma _n)_{n\in \mathbb {N}_0}$ is a given bounded sequence with $0<\sigma _{n+1}<\sigma _n$ $\forall n\in {\mathbb {N}}_0$ and $\lim _{n\rightarrow \infty }\sigma _n=0$. Thus, ${\mathcal {R}}e_n=\sigma _{|n|} e_{n+1}$.

${\mathcal {R}}$ is injective and compact, with $\mathrm {ran}\,{\mathcal {R}}$ dense in ${\mathcal {H}}$ and norm $\Vert {\mathcal {R}}\Vert _{\mathrm {op}}=\sigma _0$. (A.7) gives the singular value decomposition. The adjoint of ${\mathcal {R}}$ is

$$\begin{aligned} {\mathcal {R}}^*\;=\;{\mathcal {L}}\;=\;\sum _{n\in {\mathbb {Z}}} \sigma _{|n|}\,|e_n\rangle \langle e_{n+1}|. \end{aligned}$$

(A.8)

Thus, ${\mathcal {L}}{\mathcal {R}}=M^{(\sigma ^2)}={\mathcal {R}}{\mathcal {L}}$.

The ‘inverse of ${\mathcal {R}}$ on its range’ is the densely defined, surjective, unbounded operator ${\mathcal {R}}^{-1}:\mathrm {ran}\,{\mathcal {R}}\rightarrow {\mathcal {H}}$ acting as

$$\begin{aligned} {\mathcal {R}}^{-1}\;=\;\sum _{n\in {\mathbb {Z}}}\,\frac{1}{\sigma _{|n|}}\,|e_n \rangle \langle e_{n+1}| \end{aligned}$$

(A.9)

as a series that converges on $\mathrm {ran}\,{\mathcal {R}}$ in the strong operator sense.

1.5 A.5. The Volterra operator on $L^2[0,1]$

This is the operator $V:L^2[0,1]\rightarrow L^2[0,1]$ defined by

$$\begin{aligned} (Vf)(x)\;=\;\int _0^x \!f(y)\,\mathrm {d}y,\qquad x\in [0,1]. \end{aligned}$$

(A.10)

V is compact and injective with spectrum $\sigma (V)=\{0\}$ (thus, the spectral point 0 is not an eigenvalue) and norm $\Vert V\Vert _{\mathrm {op}}=\frac{2}{\pi }$. It’s adjoint $V^*$ acts as

$$\begin{aligned} (V^*f)(x)\;=\;\int _x^1 \!f(y)\,\mathrm {d}y,\qquad x\in [0,1], \end{aligned}$$

(A.11)

therefore $V+V^*$ is the rank-one orthogonal projection

$$\begin{aligned} V+V^*\;=\;|{\mathbf {1}}\rangle \langle {\mathbf {1}}| \end{aligned}$$

(A.12)

onto the function ${\mathbf {1}}(x)=1$.

The singular value decomposition of V is

$$\begin{aligned} V\;=\;\sum _{n=0}^\infty \sigma _n|\psi _n\rangle \langle \varphi _n|,\qquad \quad \begin{array}{rl} \sigma _n\;=&{}\!\frac{2}{(2n+1)\pi } \\ \varphi _n(x)\;=&{}\!\sqrt{2}\,\cos \frac{(2n+1)\pi }{2}x \\ \psi _n(x)\;=&{}\!\sqrt{2}\,\sin \frac{(2n+1)\pi }{2}x, \end{array} \end{aligned}$$

(A.13)

where both $(\varphi _n)_{n\in {\mathbb {N}}_0}$ and $(\psi _n)_{n\in {\mathbb {N}}_0}$ are orthonormal bases of $L^2[0,1]$.

Thus, $\mathrm {ran} V$ is dense, but strictly contained in ${\mathcal {H}}$: for example, ${\mathbf {1}}\notin \mathrm {ran}V$. (Observe, though, that the dense subspace of the polynomials on [0, 1] is mapped by V onto the non-dense $\mathrm {span}\{x,x^2,x^3,\dots \}$.)

In fact, V is invertible on its range, but does not have (everywhere defined) bounded inverse; yet $V-z\mathbb {1}$ does, for any $z\in {\mathbb {C}}{\setminus }\{0\}$ (recall that $\sigma (V)=\{0\}$), and

$$\begin{aligned} (z\mathbb {1}-V)^{-1}\psi \;=\;z^{-1}\psi +z^{-2}\!\int _0^x e^{\frac{x-y}{z}}\,\psi (y)\,\mathrm {d}y\qquad \forall \psi \in {\mathcal {H}},\; z\in {\mathbb {C}}{\setminus }\{0\}.\nonumber \\ \end{aligned}$$

(A.14)

The explicit action of the powers of V is

$$\begin{aligned} (V^n f)(x)\;=\;\frac{1}{\,(n-1)!}\int _0^x(x-y)^{n-1}f(y)\,\mathrm {d}y,\qquad n\in {\mathbb {N}}. \end{aligned}$$

(A.15)

1.6 A.6. The multiplication operator over $\Omega \subset {\mathbb {C}}$ in $L^2(\Omega )$

This is the operator $M:L^2(\Omega )\rightarrow L^2(\Omega )$, $f\mapsto zf$, where $\Omega$ is a bounded open region in ${\mathbb {C}}$. $M_z$ is a normal bounded bijection with norm $\Vert M_z\Vert _{\mathrm {op}}=\sup _{z\in \Omega }|z|$, spectrum $\sigma (M_z)={\overline{\Omega }}$, and adjoint given by $M_z^*f={\overline{z}}f$.