bibkey: hansen2013fasttrack
authors: Frank Hansen
year: 2013
title: The fast track to Loewner's theorem
doi: 10.1016/j.laa.2013.01.022
url: https://arxiv.org/abs/1112.0098
claim: A block sandwich shows that operator monotonicity on a doubled space yields operator concavity on the original space.
strata_touched:
  - D5/S3/Quantum/Lieb
license: citation-only
triage: anchor

Monotonicity to concavity through a block sandwich

Frank Hansen, The fast track to Loewner’s theorem, Linear Algebra and its Applications 438 (2013), 4557–4571, DOI https://doi.org/10.1016/j.laa.2013.01.022. The opened source is arXiv:1112.0098v3, Theorem 2.1 and its proof. No mathematical literature novelty is claimed.

The local theorem D5.S3.Quantum.Lieb.MonotoneBlockConcavity.monotone_double_space_concavity retains continuity on the positive half-line as an explicit hypothesis. Its operators are actual bounded operators on an arbitrary complete nontrivial complex Hilbert space H. Monotonicity is required on H + H; concavity is proved on H for strictly positive inputs and every real weight in [0,1]. There is no finite-dimensionality or finite-spectrum restriction.

The proof rotates diag(A,B), dominates its off-diagonal block by a positive square after adding epsilon I to the first corner and a sufficiently large scalar to the second corner, applies the operator-monotonicity premise, compresses to the first corner, and removes epsilon in operator norm through continuous functional calculus. This is a new local formal proof assembled from the existing block-operator and Mathlib CFC APIs, rather than a Lean source transplant.

Hansen also removes the continuity assumption by excluding scalar jumps. That extra argument, the Bendat–Sherman step and the existence and uniqueness of a Loewner representing measure are not provided by this module.

Verified locator

• DOI: https://doi.org/10.1016/j.laa.2013.01.022
• Source: https://arxiv.org/abs/1112.0098
• Opened version: https://arxiv.org/abs/1112.0098v3, Theorem 2.1 and proof, pages 3–5.

The bibliographic DOI was verified against the live Crossref record. The opened arXiv version is preserved in the persistent research source directory.