We discuss the antisymmetry of
certain algebras associated with a bounded linear operator on a Hilbert space ℋ. An
algebra of operators on ℋ is said to be antisymmetric if the only self-adjoint
operators it contains are multiples of the identity. If T is a bounded operator on ℋ,
let 𝒜u(T) be the norm closure of {p(T) : p is a polynomial} and let ℛu(T) be the
norm closure of {f(T) : f is a rational function with poles off the spectrum of T}.
Suppose T = T1⊕ T2⊕⋯ and ∥Tj∥→ 0 as j →∞. For a subset J of N, the
natural numbers, let TJ= ⊕{Tj: j ∈ J}. It is proved here that if 𝒜u(Tj) is
antisymmetric for each j, then 𝒜u(T) is antisymmetric if and only if for each finite
set J the polynomially convex hulls of σ(TJ) and σ(TN|J) have nonempty
intersection.
