Abstract
In this paper we initiate a study of covariance and variance for two operators on a Hilbert space, proving that the c-v (covariance-variance) inequality holds, which is equivalent to the Cauchy-Schwarz inequality. As for applications of the c-v inequality we prove uniformly the Bernstein-type inequalities and equalities, and show the generalized Heinz-Kato-Furuta-type inequalities and equalities, from which a generalization and sharpening of Reid's inequality is obtained. We show that every operator can be expressed as a p-hyponormal-type, and a hyponormal-type operator. Finally, some new characterizations of the Furuta inequality are given.
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Lin, C.S. On Variance and Covariance for Bounded Linear Operators. Acta Math Sinica 17, 657–668 (2001). https://doi.org/10.1007/s101140100133
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DOI: https://doi.org/10.1007/s101140100133
Keywords
- Covariance-variance inequality
- Bernstein inequality
- Reid's inequality
- Furuta inequality
- Löwner-Heinz formula