Invariant Subspaces of Completely Continuous Operators

1954/09/01 by N. Aronszajn, K. T. Smith · 5 citations
Mathematics · #Holomorphic and Operator Theory #Advanced Banach Space Theory #Mathematical Analysis and Transform Methods #Mathematics #Linear subspace #Invariant (physics) #Reflexive operator algebra #Pure mathematics #Invariant subspace #Invariant subspace problem #Compact operator #Unbounded operator #Mathematical physics #Banach space #Finite-rank operator

paper · doi:10.2307/1969637

openalex publication_date 1954/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

Abstract : A proof is presented of the theorem that if B is a Banach space and if T is a completely continuous operator in B, there then exist proper invariant subspaces of T. The proof assumes strong convergence, complete continuity in the sense that any bounded set is transformed by T into a set with compact closure, and a strictly convex norm. The same theorem is proved for a Hilbert space; the theorem utilizes the concepts of weak and strong convergenece of elements and operators. The simplifying feature of the latter proof is that the metric propjections coincide with the usual orthogonal projections.

Citations

• On two problems concerning linear transformations in hilbert space

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• Invariant subspace problem [wikipedia]