Abstract
In 1982, S. Friedland proved that a bounded linear operator A on a Hilbert space is normal if and only if (αI + A + A*)2 ≧ AA* − A*A ≧ −(αI + A + A*)2 for all real α. And he conjectured the inequality (αI + A + A*)2 ≧ AA* − A*A for all real α will imply that A*A − AA* ≧ 0, i.e., A is hyponormal. But his conjecture is incorrect. In this note I’ll give a counter-example for his conjecture.
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References
S. Friedland,A characterization of normal operators, Isr. J. Math.42 (1982), 235–240.
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Yoshino, T. A counter-example to a conjecture of Friedland. Israel J. Math. 75, 273–275 (1991). https://doi.org/10.1007/BF02776028
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DOI: https://doi.org/10.1007/BF02776028