an: Zbl 1186.47012
au: Fujii, Masatoshi; Zuo, Hongliang
ti: Matrix order in Bohr inequality for operators.
la: EN
so: Banach J. Math. Anal. 4, No. 1, 21-27, electronic only (2010).
py: 2010
dt: J
cc: *47A63 47B15
ut: Bohr inequality for operators; matrix order; parallelogram law for
    operators; absolute value of operators
ab: Summary: The classical Bohr inequality says that $|a+b|^2\le p|a|2+q|b|^2$
    for all scalars $a,b$ and $p,q>0$ with $\frac1p+ \frac1q=1$. The equality
    holds if and only if $(p-1)a=b$. Several authors discussed operator versions
    of the Bohr inequality. In this paper, we give a unified proof to operator
    generalizations of the Bohr inequality. One viewpoint of ours is a matrix
    inequality, and the other is a generalized parallelogram law for the
    absolute value of operators, i.e., for operators $A$ and $B$ on a Hilbert
    space and $t\ne 0$,
 $$|A-B|^2+ \tfrac 1t|tA+B|^2= (1+t)|A|^2+
    \big(1+\tfrac1t\big) |B|^2.$$