On the joint spectra of operators and antiunitaries

Abstract

Camara and Krejˇciˇr´ık have studied properties of operators concerning with an
antiunitary operator C (see [1]). In this paper we show that if T = (T1, ..., Tn)
is a commuting n-tuple of Hilbert space operators and C is an antinuitary, then
σ(CTC−1) = σ(T)∗, where σ(T) is the Taylor spectrum of T, σ(T)∗ = { z =
(z1, ..., zn) : z = (z1, ..., zn) ∈ σ(T)} and CTC−1 = (CT1C−1, ...,CTnC−1). Also
we will show σX(CTC−1) = {z : z ∈ σX(T)}, where σX(T) is the Xia spectrum of
T.