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On spectra of Lüders operations. (English) Zbl 1153.81410
Summary: We show that all the eigenvalues of certain generalized Lüders operations are non-negative real numbers in two cases of interest. In particular, given a commuting $$n$$-tuple $$\mathcal A=(A_1,\dots,A_n)$$ consisting of positive operators on a Hilbert space $$\mathcal H$$, satisfying $$\sum_{j=1}^nA_j=I$$, we show that the spectrum of the Lüders operation: $\Lambda_{\mathcal A}: \mathcal B(\mathcal H)\ni X\mapsto \sum_{j=1}^nA_j^{1/2}XA_j^{1/2}\in \mathcal B(\mathcal H)$ is contained in $$[0,\infty)$$, so the only solution of the equation $$\Lambda(X)=I-X$$ is the “expected” one: $$X=\tfrac 12 I$$.

##### MSC:
 81R15 Operator algebra methods applied to problems in quantum theory 47N50 Applications of operator theory in the physical sciences 81P15 Quantum measurement theory, state operations, state preparations 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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##### References:
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