An operator equality involving a continuous field of operators and its norm inequalities. (English) Zbl 1156.47021

Using Th. M. Rassias’ result [Bull. Sci. Math., II. Sér. 108, 95–99 (1984; Zbl 0544.46016)], a normed space \(X\) is an inner product space (i.p.s) if and only if, for any finite set of vectors \(x_1,\dots, x_n\) in \(X\),
\[ \sum_{i=1}^n\left\|x_i-\frac1n \sum_{j=1}^n x_j\right\|^2=\sum \left\|x_i\right\|^2-n\left\|\frac1n \sum_{j=1}^n x_j\right\|^2 .\tag{1} \]
This equality is of fundamental importance in the study of normed spaces and inner product spaces.
In this article, the authors establish an operator version of equality (1) involving continuous fields of operators and integral means of operators
\[ \int_T \left|A_t-\int_T A_s \,dP(s)\right|^2 \,dP(t)=\int |A_t|^2 \,dP(t)-\left|\int_T A_t \,dP(t)\right|^2. \]
Moreover, the authors present some inequalities concerning various norms such as Schatten \(p\)-norms that form natural generalizations of the identity [cf. O. Hirzallah, F. Kittaneh and M. S. Moslehian, “Schatten \(p\)-norm inequalities related to a characterization of inner product spaces”, Math. Inequal. Appl. (to appear; arxiv:0801.2726v1)] and mention that the inequalities related to Schatten \(p\)-norms are useful to operator theory and mathematical physics.


47A62 Equations involving linear operators, with operator unknowns
46C15 Characterizations of Hilbert spaces
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A24 Matrix equations and identities


Zbl 0544.46016
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