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- W1884174937 abstract "The variance of a bounded linear operator $a$ on a Hilbert space $H$ at a unit vector $h$ is defined by $D_h(a)=|ah|^2-| |^2$. We show that two operators $a$ and $b$ have the same variance at all vectors $hin H$ if and only if there exist scalars $sigma,lambda$ with $|sigma|=1$ such that $b=sigma a+lambda1$ or $a$ is normal and $b=sigma a^*+lambda1$. Further, if $a$ is normal, then the inequality $D_h(b)leqkappa D_h(a)$ holds for some constant $kappa$ and all unit vectors $h$ if and only if $b=f(a)$ for a Lipschitz function $f$ on the spectrum of $a$. Variants of these results for C$^*$-algebras are also proved. We also study the related, but more restrictive inequalities $|bx-xb|leq |ax-xa|$ supposed to hold for all $xin B(H)$ or for all $xin B(H^n)$ and all positive integers $n$. We consider the connection between such inequalities and the range inclusion $d_b(B(H))subseteq d_a(B(H))$, where $d_a$ and $d_b$ are the derivations on $B(H)$ induced by $a$ and $b$. If $a$ is subnormal, we study these conditions in particular in the case when $b$ is of the form $b=f(a)$ for a function $f$." @default.
- W1884174937 created "2016-06-24" @default.
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- W1884174937 date "2013-02-08" @default.
- W1884174937 modified "2023-09-27" @default.
- W1884174937 title "Variance of operators and derivations" @default.
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