Theoretical analysis reveals necessary and sufficient criteria for positive inverse operators on C*-algebras, indicating that positivity fails even when spectra lie on the unit circle.
For a positive and invertible linear operator T acting on a C^*-algebra, we give necessary and sufficient criteria for the inverse operator T⁻¹ to be positive, too. Moreover, a simple counterexample shows that T⁻¹ need not be positive even if T is unital and its spectrum is contained in the unit circle.
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Glück et al. (2024) studied this question.