Note on a paper by C. C. Brown

Abstract

Let H be a real or complex Hilbert space and let L p(1≦p<∞) denote the set of compact self adjoint operators J whose sequence (λ n) of non-zero eigenvalues satisfies $$\left\| J \right\|_p = (\sum {\left| {\lambda _n } \right|^p } )^{1/p} < \infty .$$ In the paper referred to in the title C. C. Brown showed that the map J → ¦J¦ is continuous on L 1 relative to ∥·∥1 and, for a sequence (A n) of non-negative elements of L 2, ∥A n-A∥2→0 if and only if ∥A n 2 -A 2∥1→0. This note simplifies Brown's methods considerably and proves that J → ¦J is continuous on L p relative to ∥·||p and that for a sequence (A n) of non-negative elements of L p, ∥A n-A∥ p →0 if and only if ∥A n p -Ap∥1 → 0.