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Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.<\/jats:p>","DOI":"10.1007\/s00211-023-01382-8","type":"journal-article","created":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T23:02:24Z","timestamp":1703718144000},"page":"1-38","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates"],"prefix":"10.1007","volume":"156","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sophie","family":"Puttkammer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,12,27]]},"reference":[{"key":"1382_CR1","doi-asserted-by":"crossref","unstructured":"Agmon, S.: Lectures on Elliptic Boundary Value Problems. 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