Vol. 89, No. 2, 1980

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Singly generated antisymmetric operator algebras

John Bligh Conway and Wacław Szymański

Vol. 89 (1980), No. 2, 269–277

We discuss the antisymmetry of certain algebras associated with a bounded linear operator on a Hilbert space . An algebra of operators on 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 , let 𝒜u(T) be the norm closure of {p(T) : p is a polynomial} and let u(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.

Mathematical Subject Classification
Primary: 47D25, 47D25
Received: 25 September 1978
Revised: 11 May 1979
Published: 1 August 1980
John Bligh Conway
Wacław Szymański