Let $ H^{N} $ be a $ N $-dimensional Hilbert space, $ S, T\in L(H^{N}). $ Suppose that $ T $ is invertible and positive operator $(<Tx,x>\geq0 ) $ such that $ ST=TS. $ I can't prove that $ ST^{r}=T^{r}S$, for ay real number $ r\in \mathbb {R}$. We know since $ T $ is positive and invertible, then $ T^{r} $ is well-defined for any real number $ r\in \mathbb {R}$.
2026-02-24 11:58:15.1771934295
Commutant of positive invertible operator
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