Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos

1966/03/01 by Allen R. Bernstein, Abraham Robinson · 2 citations
Mathematics · #Holomorphic and Operator Theory #Algebraic and Geometric Analysis #Spectral Theory in Mathematical Physics #Mathematics #Subspace topology #Invariant subspace #Invariant (physics) #Combinatorics #Pure mathematics #Mathematical physics #Mathematical analysis #Linear subspace

paper · pdf · doi:10.2140/pjm.1966.16.421

openalex publication_date 1966/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

The following theorem is proved. Let T be a bounded linear operator on an infinite-dimensional Hubert space H over the complex numbers and let p(z) 0 be a polynomial with complex coefficients such that p(T) is completely continuous (compact). Then T leaves invariant at least one closed linear subspace of H other than H or 0.

Citations

• Invariant Subspaces of Completely Continuous Operators
• Invariant subspaces of polynomially compact operators
• Completeness in the theory of types

Cited by

• The invariant subspace problem
• Invariant subspace problem [wikipedia]