Pith Number

pith:RWZC34G3

pith:2026:RWZC34G3WGGWZKHMOGN4GZVGI5

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A proof of Esterle's conjecture on negative powers of Hilbert-space contractions

Thomas Ransford

For any closed zero-measure set E on the unit circle, there is a growth sequence u_n such that a contraction with spectrum in E and ||T^{-n}||=O(u_n) must be unitary.

arxiv:2605.16004 v1 · 2026-05-15 · math.FA · math.CV

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Claims

C1strongest claim

For each closed subset E of the unit circle of Lebesgue measure zero, there exists a positive sequence u_n→∞ with the following property: if T is a contraction on a Hilbert space such that σ(T)⊂E and ||T^{-n}||=O(u_n) as n→∞, then T is a unitary operator.

C2weakest assumption

The generalization that closed subsets E of Lebesgue measure zero remain removable for certain unbounded holomorphic functions of moderate growth near E (with the notion of moderate growth depending on E). This is the key tool invoked to handle the operator-theoretic conclusion, as described in the abstract.

C3one line summary

Proves that for every closed zero-measure subset E of the unit circle there exists u_n to infinity making contractions T with σ(T) ⊂ E and ||T^{-n}|| = O(u_n) unitary.

References

5 extracted · 5 resolved · 0 Pith anchors

[1] J. Esterle. Distributions on Kronecker sets, strong forms of uniqueness, and closed ideals ofA +.J. Reine Angew. Math., 450:43–82, 1994 1994

[2] J. Esterle. Uniqueness, strong forms of uniqueness and negative powers of contractions. InFunctional analysis and operator theory (Warsaw, 1992), volume 30 ofBanach Center Publ., pages 127–145. Polish 1992

[3] K. Kellay. Contractions et hyperdistributions ` a spectre de Carleson.J. London Math. Soc. (2), 58(1):185–196, 1998 1998

[4] T. Ransford. Negative powers of Hilbert-space contractions.J. Funct. Anal., 286(10):Paper No. 110397, 21, 2024 2024

[5] M. Zarrabi. Contractions ` a spectre d´ enombrable et propri´ et´ es d’unicit´ e des ferm´ es d´ enombrables du cercle.Ann. Inst. Fourier (Grenoble), 43(1):251–263, 1993. D´epartement de math´ematique 1993

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:01:48.550812Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8db22df0dbb18d6ca8ec719bc366a6475c60cae7a46e5f94d17758f171fe2f65

Aliases

arxiv: 2605.16004 · arxiv_version: 2605.16004v1 · doi: 10.48550/arxiv.2605.16004 · pith_short_12: RWZC34G3WGGW · pith_short_16: RWZC34G3WGGWZKHM · pith_short_8: RWZC34G3

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/RWZC34G3WGGWZKHMOGN4GZVGI5 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 8db22df0dbb18d6ca8ec719bc366a6475c60cae7a46e5f94d17758f171fe2f65

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "25005ce54d2d62930734fd851f18e028eabc05c2d7162dd4ae8187952803cdb1",
    "cross_cats_sorted": [
      "math.CV"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.FA",
    "submitted_at": "2026-05-15T14:35:33Z",
    "title_canon_sha256": "b4e2d5870f6c4c4925033666fbb3d3414172f6ae3048b39e096c7e74b790be8e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16004",
    "kind": "arxiv",
    "version": 1
  }
}