Abstract
Starting from the fact that, for projectionsP andQ in Hilbert space, equality ofPQ andQP (i.e., commutativity) is equivalent to the equality of the triple productsPQP andQPQ, the spectral resolutions ofPQP andQPQ for not necessarily commuting projections are compared. It is shown that the respective eigenspaces are isometric and display a curious biorthogonality to be described below. A more general setup relates spectral properties of operatorsTT* andT*T for boundedT. The result is connected with Mittelstaedt's theory of a probability theory for quantum mechanics.
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References
Bachman, G., and Narici, L. (1966).Functional Analysis. Academic Press, New York.
Hirzebruch, F., and Scharlau, W. (1971).Einführung in die Funktionalanalysis. B. I. Hochschultaschenbücher 296a, Mannheim.
Jauch, J. M. (1973).Foundations of Quantum Mechanics. Addison-Wesley, Reading, Massachusetts.
Mittelstaedt, P. (1976).Philosophische Probleme der modernen Physik. B. I. Hochschultaschenbücher 50, Mannheim.
Radjavi, H., and Rosenthal, P. (1973).Invariant Subspaces. Springer, Berlin.
Rudin, W. (1973).Functional Analysis. McGraw-Hill, New York.
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Rehder, W. Spectral properties of products of projections in quantum probability theory. Int J Theor Phys 18, 791–805 (1979). https://doi.org/10.1007/BF00670458
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DOI: https://doi.org/10.1007/BF00670458