On the joint spectra of operators and antiunitaries

Key Points

The antiunitary operator transforms the Taylor spectrum of commuting operators, changing its properties significantly.
The main finding shows that the spectrum of the transformed operator matches the conjugate of the original operator's spectrum.
Analysis reveals the identical relationship for the Xia spectrum under the action of the antiunitary operator.
These findings may suggest deeper connections in spectral theory and potentially influence future studies in quantum mechanics.

Abstract

Camara and Krejci? ? k 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 antiunitary, then (CTC^-1) = (T) ^, where (T) is the Taylor spectrum of T, (T) ^ = \z = (z₁, , zₙ): z = (z₁, , zₙ) (T) \ and CTC^-1 = (CT₁C^-1, , CTₙC^-1). Also we will show ₓ (CTC^-1) = \z: z ₓ (T) \, where ₓ (T) is the Xia spectrum of T.

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Cite This Study

Chō et al. (Wed,) studied this question.

synapsesocial.com/papers/69a75c1ac6e9836116a24952 https://doi.org/https://doi.org/10.2298/fil2515017c

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