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Distances between operators acting on different Hilbert spaces

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Pith's one-line read This paper proves that three natural distances between operators on different Hilbert spaces are quantitatively equivalent, with unitary and isometric distances exactly equal for self-adjoint operators whose essential spectrum contains 0.

desk verdict Solid self-adjoint results, but the general version of Theorem C rests on a bad norm identity that the authors need to repair or retract. read the letter →

arxiv 2412.13165 v2 pith:FSTWF53H submitted 2024-12-17 math.SP

Olaf Post , Sebastian Zimmer This is my paper · ORCID
classification math.SP MSC 47A1047A5847B1547A55
keywords unitarydistanceisometricquasi-unitaryequivalencecrudemultiplicityfunctionsLévy-ProkhorovspectraloperatorconvergencevaryingHilbertspaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Operators in applications often live on different Hilbert spaces—think of a Laplacian on a fractal approximated by discrete graph Laplacians, or a manifold squeezed to a graph. This paper shows that three natural ways of measuring how close two such operators are carry the same quantitative information. For self-adjoint operators with 0 in the essential spectrum, the distance between unitary orbits and the distance obtained by embedding both spaces isometrically into a common space are exactly equal, and both equal a spectral distance built from Hausdorff distance plus multiplicity-aware counting of discrete eigenvalues. For arbitrary bounded operators, a third distance based on quasi-unitary identification operators is equivalent to the isometric distance up to the universal factor $\sqrt{3}$. Consequently, convergence of operators on varying spaces can be measured by any of these metrics with the same convergence speed, and spectral convergence becomes a special case of metric convergence.

What carries the argument

The load-bearing objects are the crude multiplicity function $\alpha_R(\lambda)=\lim_{\varepsilon\to 0}\operatorname{rank} \mathbf{1}_{(\lambda-\varepsilon,\lambda+\varepsilon)}(R)$, which records how many spectral directions accumulate at each point, together with the Lévy-Prokhorov distance between such functions; the isometric embedding picture with difference operator $D=\iota_1R_1\iota_1^* - \iota_2R_2\iota_2^*$; and the quasi-unitary identification operator $J:H_1\to H_2$ with defect terms $\|R_1^*(\mathrm{id}-J^*J)R_1\|^{1/2}$, $\|R_2^*(\mathrm{id}-JJ^*)R_2\|^{1/2}$, $\|JR_1-R_2J\|$, and $\|JR_1^*-R_2^*J\|$. The proof of Theorem B uses invariance of $\alpha_R$ under isometries when $0$ is in the essential spectrum; the proof of Theorem C constructs a parent space $H_1\oplus H_2$, sets $\iota_1=((\mathrm{id}-J^*J)^{1/2},J)$, $\iota_2=(0,\mathrm{id})$, and bounds $D$ by decomposing it as $P_2DP_1+P_2DP_1^\perp+P_2^\perp DP_1$ with orthogonal projections onto the ranges of the isometries, yielding the constant $\sqrt{3}$ from Pythagoras.

What would settle it

For $R_1=1$ on $\mathbb{C}$, $R_2=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$ on $\mathbb{C}^2$, and $J=\mathrm{diag}(1/\sqrt{2},0)$, compute the two numbers $\|R_2^*(\mathrm{id}-JJ^*)R_2\|$ and $\|R_2(\mathrm{id}-JJ^*)R_2^*\|$; they come out as $1/2$ and $1$, respectively, so the norm identity used in the proof of Theorem C fails. Computing $d_{\mathrm{iso}}$ and $d_{\mathrm{que}}$ exactly for this pair then settles whether the universal $\sqrt{3}$ bound itself survives for non-normal operators or needs an adjoint-adjusted distance.

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Extended reading notes

Core claim

The paper's central claim is that the choice of how to compare operators on different Hilbert spaces does not matter quantitatively. Concretely, if $R_1$ and $R_2$ are bounded self-adjoint operators and $0$ lies in the essential spectrum of both, then $d_{\mathrm{uni}}(R_1,R_2)=d_{\mathrm{iso}}(R_1,R_2)=d_{\mathrm{spec}}(R_1,R_2)$; and for any bounded operators on separable Hilbert spaces, $d_{\mathrm{que}}(R_1,R_2)\le d_{\mathrm{iso}}(R_1,R_2)\le \sqrt{3}\,d_{\mathrm{que}}(R_1,R_2)$. The equality is driven by the fact that $0$ in the essential spectrum makes the crude multiplicity function invariant under isometric embedding, so unitary-orbit data survive the embedding. The equivalence with the quasi-unitary distance is proved by constructing an explicit parent space $H_1\oplus H_2$ with an embedding built from the defect operator of the identification $J$, and estimating the difference operator by a Pythagorean split of its three off-diagonal pieces.

Load-bearing premise

The universal factor $\sqrt{3}$ bound between quasi-unitary and isometric distances for general bounded operators rests on the assumption that the defect of the identification operator has the same size whether the operator sits on the left or on the right of the defect factor; this is automatic for self-adjoint operators but not for arbitrary bounded ones.

Editorial extensions

If this is right

  • For self-adjoint operators with $0$ in the essential spectrum, the three metrics $d_{\mathrm{uni}}$, $d_{\mathrm{iso}}$, and $d_{\mathrm{spec}}$ give identical distances; convergence in any one of them is convergence in all, with the same speed.
  • For any bounded pair, $d_{\mathrm{que}}$ and $d_{\mathrm{iso}}$ are equivalent up to the universal constant $\sqrt{3}$, so quasi-unitary convergence and generalized norm resolvent convergence are equivalent even across different parent spaces.
  • The unitary distance $d_{\mathrm{uni}}(R_1,R_2)$ always bounds the Hausdorff distance of the spectra; when both operators have purely essential spectrum, the two distances are exactly equal.
  • If $0$ is missing from an essential spectrum, the equality $d_{\mathrm{uni}}=d_{\mathrm{iso}}$ can fail, as the counterexamples in Section 6 show, so the spectral condition is genuinely needed.
  • The quotient of self-adjoint operators with $0$ in the essential spectrum by approximate unitary equivalence is a Hausdorff metric space isometric to the space of crude multiplicity functions with the Lévy-Prokhorov distance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The constant $\sqrt{3}$ in Theorem C is probably not optimal: the paper itself exhibits a lower bound of $\sqrt{2}$, and tightening the gap between $\sqrt{2}$ and $\sqrt{3}$ would sharpen all convergence-speed statements for the quasi-unitary distance.
  • Because the proof of the bounded-operator upper bound compares two quadratic defect expressions that need not agree for non-normal operators, a natural repair is to redefine or adjust the quasi-unitary distance by placing adjoints symmetrically; the self-adjoint applications in spectral geometry and fractal approximation would be unaffected.
  • The metric unification suggests a practical recipe for numerical analysis on varying spaces: pick whichever of the three distances is easiest to compute in a given problem, and the resulting convergence criterion (and even its order) transfers to the others.
  • The isometric distance is formally a Gromov-Hausdorff-type distance for graphs of operators with isometries constrained to diagonal form; exploring that connection could yield a Banach-space analogue or a link to the classical gap distance between subspaces.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper introduces and compares three distances between bounded operators acting on different separable Hilbert spaces: the unitary distance duni, the isometric distance diso (after Weidmann), and a quasi-unitary distance dque with sandwiched defect terms. The main theorems assert that for self-adjoint operators with 0 in the essential spectrum, duni equals diso and both equal the spectral distance dspec (Theorems A and B, Corollary D); that for arbitrary bounded operators, dque and diso are equivalent up to the universal constant sqrt(3) (Theorem C); and that the associated convergence notions for resolvents—Weidmann convergence, QUE-convergence, and convergence in these distances—coincide with matching speeds (Theorem F). The proofs rely on crude multiplicity functions and the Azoff–Davis/Davidson equality for unitary orbits, plus explicit embedding constructions for the isometric and quasi-unitary distances.

Significance. If established, the framework gives a coherent quantitative language for spectral approximation of operators on varying Hilbert spaces, unifying earlier convergence notions and showing that spectral, unitary, isometric, and quasi-unitary distances carry the same information (up to a constant). The paper is careful in attributing the unitary-orbit results to Azoff–Davis and Davidson, and the self-adjoint equality Theorem B appears sound and is proven with a clean crude-multiplicity argument. The self-adjoint consequences (Corollary D, Theorem A, and the self-adjoint parts of Theorem F) are valuable. However, the general bounded-operator part of Theorem C is not proven as written, and this gap propagates into the non-self-adjoint statements of Corollary E and Theorem F.

major comments (2)
  1. [Section 5, Theorem C proof, Eq. (5.7)] The upper-bound proof of Theorem C for general bounded operators is not supported by Proposition 5.4. The estimate in Eq. (5.7) bounds ‖P2DP⊥1‖² by ‖R2*(id−JJ*)R2‖, but the identity proved in the paper gives ‖P2DP⊥1‖² = ‖R2(id−JJ*)R2*‖ (Eq. (5.3j)); Eq. (5.3h), which concerns the different block P⊥1DP2, is itself misstated for non-self-adjoint R2 (the correct right-hand side is ‖R2*(id−JJ*)R2‖). These two numbers differ in general: for R2 = [[0,1],[0,0]] and J = (1/√2)(1,0) one has ‖R2*(id−JJ*)R2‖ = 1/2 but ‖R2(id−JJ*)R2*‖ = 1. Since δJ controls only the former, the chain (2δ²+δ²) in (5.7) is unjustified for non-self-adjoint operators. The self-adjoint case is unaffected, and the first inequality dque ≤ diso can be repaired using Eq. (5.3e), but as written Theorem C is not established in the generality stated.
  2. [Section 7, Definition 7.2 and Theorem F] There is a mismatch between the QUE-convergence conditions and the quasi-unitary distance used in the equivalence. Definition 7.2 imposes estimates on ‖Rn(id−J*J)Rn‖ and ‖R∞(id−JJ*)R∞‖ (no adjoints on the resolvents), while the distance dque in Eq. (1.10b) uses ‖R1*(id−J*J)R1‖ and ‖R2*(id−JJ*)R2‖. For non-self-adjoint resolvents these are not the same expressions. Consequently the asserted equivalence (c)⇔(d) in Theorem F for general closed operators does not follow from Theorem C without an additional argument or a revision of Definition 7.2. The self-adjoint part of Theorem F, where the resolvents are self-adjoint, is not affected.
minor comments (3)
  1. [Section 6.2, Proposition 6.3] The computation of δJ(R,0) drops the fourth term ‖JR*‖ from Eq. (1.10b); the stated value ‖R‖/√2 is still correct (one may take J = (1/√2)U with U unitary and use ‖SR‖²+‖CR‖² = ‖R‖²), but the proof should account for this term explicitly.
  2. [Section 5, Proposition 5.4] Eq. (5.3h) should be corrected to ‖P⊥1DP2‖² = ‖R2*(id−JJ*)R2‖; this typo is related to the gap in Theorem C and should be fixed even if the main theorem is ultimately repaired by a different argument.
  3. [Section 1.1, Eq. (1.4)] The notation in the definition of ddisc is hard to read because the spectral projection symbol appears as /BD; the authors should ensure the typeset version uses a clear projection notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main comparison theorems are derived from self-contained definitions and externally attributed results, not from hidden inputs.

full rationale

The paper's central results compare three independently defined distances. Theorem B is proved by reducing through Azoff-Davis's theorem (Theorem 3.3, based on [AD84] and [D86]) and Lemma 4.4 on crude multiplicity functions under isometries; the crude multiplicity formalism is developed self-containedly in Section 2. Theorem C is proved from Proposition 5.4, whose identities are derived directly from J = ι2*ι1 and the block decomposition of D = ι1R1ι1* − ι2R2ι2*. No term in δJ is fitted or defined as a disguised version of diso; the four norms in (1.10b) are independent quantities and the proof bounds them using the explicit isometric embedding (5.6). The quasi-unitary distance is a modification of the first author's earlier notion, but the paper defines it explicitly and proves the comparison rather than importing the comparison from [PZ22]. Theorem F is a continuation of [PZ22], but the proof is repeated in Section 7 and the equivalences follow from Theorems A–C. The cited Azoff–Davis and Davidson results are independent external theorems; self-citations to [P06], [PS20a], and [PZ22] are contextual and not used to force the main inequalities. A possible defect in the upper-bound proof of Theorem C, concerning the off-diagonal norm identity in Eq. (5.7), is a correctness or technical-gap issue rather than a circularity: it does not make an output equal to an input by construction. Therefore, no circular step meets the evidentiary standard required by the review rules.

Assumptions & free parameters 0 free parameters · 7 assumptions · 2 invented entities

No fitted constants or ad hoc parameters: this is pure mathematics. The load-bearing inputs are classical theorems (Azoff-Davis [AD84], Davidson [D86], Halmos [Hal69]) and standard functional analysis. The claimed new theorems depend on (i) the black-box equality duni = δ(α) from [AD84], (ii) Lemma 4.4, and (iii) the norm estimate (5.7), which is incorrect for non-normal operators in the general form of Theorem C. The distances diso and dque are novel constructs introduced by the paper, not external entities with independent evidence.

assumptions (7)
  • standard math Spectral theorem and Borel functional calculus for bounded self-adjoint operators
    Foundation for crude multiplicity functions αR and their measures, Section 2.3, Proposition 2.6.
  • domain assumption Azoff-Davis theorem [AD84, Thm. 1.3]: unitary-orbit distance equals Lévy-Prokhorov distance of crude multiplicity functions for self-adjoint operators in one space
    Used as a black box in Theorem 3.3 and Theorem B; not reproven in this paper.
  • domain assumption Davidson [D86, Prp. 1.3]: dspec = duni for self-adjoint bounded operators
    Theorem A; the paper offers a proof sketch via crude multiplicity functions but credits [D86].
  • standard math Halmos two-projections theorem [Hal69]
    Used to compute diso and dque for scalar multiples in Section 6.
  • standard math C*-norm identity ||A*A|| = ||AA*|| and norm preservation by isometries
    Core tool in Proposition 5.4 and the proof of Theorem C.
  • standard math 0 ∈ σess(R) implies rank 1_{(-ε,ε)}(R) = ∞ for every ε > 0, for self-adjoint R
    Essential for Lemma 4.4 (crude multiplicity preserved under isometric embedding) and Theorem B.
  • domain assumption Separability of all Hilbert spaces
    Keeps the crude multiplicity measure N0 ∪ {∞}-valued; stated in Section 2.1.
invented entities (2)
  • isometric distance diso (Eq. (1.8))
    purpose: Metric comparing operators acting on different Hilbert spaces via isometric embeddings into a common space
    A new definition in this paper; it is precisely defined and computed in examples (Propositions 6.1, 6.2), but it is an internal mathematical construct with no falsifiable handle outside the paper.
  • quasi-unitary distance dque with sandwiched defect terms (Eq. (1.10))
    purpose: Quantitative measure of closeness to unitary equivalence via identification operators J
    Modified from the authors' earlier QUE notion ([PZ22], [PS20a]); a definition rather than an empirical entity, with no external falsifiable handle.

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Pith. "Pith review of Distances between operators acting on different Hilbert spaces." pith.science (2026). https://pith.science/paper/FSTWF53H

@misc{pith2026241213165,
  author       = {Pith},
  title        = {Pith review of: Distances between operators acting on different Hilbert spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FSTWF53H}},
  note         = {Machine review of arXiv:2412.13165}
}
abstract

The aim of this article is to define and compare several distances (or metrics) between operators acting on different (separable) Hilbert spaces. We consider here three main cases of how to measure the distance between two bounded operators: first by taking the distance between their unitary orbits, second by isometric embeddings (this generalises a concept of Weidmann) and third by quasi-unitary equivalence (using a concept of the first author of the present article). Our main result is that the unitary and isometric distances are equal provided the operators are both self-adjoint and have $0$ in their essential spectra. Moreover, the quasi-unitary distance is equivalent (up to a universal constant) with the isometric distance for any pair of bounded operators. The unitary distance gives an upper bound on the Hausdorff distance of their spectrum. If both operators have purely essential spectrum, then the unitary distance equals the Hausdorff distance of their spectra. Using a finer spectral distance respecting multiplicity of discrete eigenvalues, this spectral distance equals the unitary distance also for operators with essential and discrete spectrum. In particular, all operator distances mentioned above are equal to this spectral distance resp. controlled by it in the quasi-unitary case for self-adjoint operators with $0$ in the essential spectrum. We also show that our results are sharp by presenting various (counter-)examples. Finally, we discuss related convergence concepts complementing results from our first article arXiv:2202.03234

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