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edit approved Jun 17, 2015 at 5:11
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edit approved Jun 17, 2015 at 5:11
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In Rota's paper (An Alternierende Verfahren for General Positive Operators), Theorem 2 says that: Let P$P$ be a doubly stochastic operator which is selfadjoint in L2 (S, \Simga, \mu)$L^2 (S, \Sigma, \mu)$. Then there is a dilation of the sequence of operators P^{2n}$P^{2n}$ into a martingale E_n$E_n$.

My question is, what does he mean by a dilation? Is he referring to the definition of dilation: Q$Q$ is a dilation of T$T$ in T^n = EQ^n Dv$T^n = EQ^n D$ where T:V\to V$T:V\to V$, Q:W\to W$Q:W\to W$ and D: V\to W$D: V\to W$?

In Rota's paper (An Alternierende Verfahren for General Positive Operators), Theorem 2 says that: Let P be a doubly stochastic operator which is selfadjoint in L2 (S, \Simga, \mu). Then there is a dilation of the sequence of operators P^{2n} into a martingale E_n.

My question is, what does he mean by a dilation? Is he referring to the definition of dilation: Q is a dilation of T in T^n = EQ^n Dv where T:V\to V, Q:W\to W and D: V\to W?

In Rota's paper (An Alternierende Verfahren for General Positive Operators), Theorem 2 says that: Let $P$ be a doubly stochastic operator which is selfadjoint in $L^2 (S, \Sigma, \mu)$. Then there is a dilation of the sequence of operators $P^{2n}$ into a martingale $E_n$.

My question is, what does he mean by a dilation? Is he referring to the definition of dilation: $Q$ is a dilation of $T$ in $T^n = EQ^n D$ where $T:V\to V$, $Q:W\to W$ and $D: V\to W$?

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Dilation of positive operators into martingales

In Rota's paper (An Alternierende Verfahren for General Positive Operators), Theorem 2 says that: Let P be a doubly stochastic operator which is selfadjoint in L2 (S, \Simga, \mu). Then there is a dilation of the sequence of operators P^{2n} into a martingale E_n.

My question is, what does he mean by a dilation? Is he referring to the definition of dilation: Q is a dilation of T in T^n = EQ^n Dv where T:V\to V, Q:W\to W and D: V\to W?