{"paper":{"title":"Direct sums of finite dimensional $SL^\\infty_n$ spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Richard Lechner","submitted_at":"2017-09-07T15:09:26Z","abstract_excerpt":"$SL^\\infty$ denotes the space of functions whose square function is in $L^\\infty$, and the subspaces $SL^\\infty_n$, $n\\in\\mathbb{N}$, are the finite dimensional building blocks of $SL^\\infty$.\n  We show that the identity operator $I_{SL^\\infty_n}$ on $SL^\\infty_n$ well factors through operators $T : SL^\\infty_N\\to SL^\\infty_N$ having large diagonal with respect to the standard Haar system. Moreover, we prove that $I_{SL^\\infty_n}$ well factors either through any given operator $T : SL^\\infty_N\\to SL^\\infty_N$, or through $I_{SL^\\infty_N}-T$. Let $X^{(r)}$ denote the direct sum $\\bigl(\\sum_{n\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02297","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}