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wlog one can assume that $||E_1||=1$.
Let $X$ be the span of the matrix units $E_{11},E_{21},E_{31}$. Then for every $2$ linearly independent operators $E_1, E_2$ in this space there exists a unitary matrix $U$ such that $UE_1=E_{11}$, $UE_2=\lambda E_{21}+E_{11}$. But $E_{11}$ and $E_{12}$ \lambda E_{21}+E_{11}$are not simultaneously singular valued decomposable. 2 added 7 characters in body wlog one can assume that$||E_1||=||E_2||=1$.||E_1||=1$.
Let $X$ be the span of the matrix units $E_{11},E_{21},E_{31}$. Then for every $2$ linearly independent operators $E_1, E_2$ in this space there exists a unitary matrix $U$ such that $UE_1=E_{11}$, $UE_2=E_{21}$. UE_2=\lambda E_{21}+E_{11}$. But$E_{11}$and$E_{12}$are not simultaneously singular valued decomposable. 1 wlog one can assume that$||E_1||=||E_2||=1$. Let$X$be the span of the matrix units$E_{11},E_{21},E_{31}$. Then for every$2$linearly independent operators$E_1, E_2$in this space there exists a unitary matrix$U$such that$UE_1=E_{11}$,$UE_2=E_{21}$. But$E_{11}$and$E_{12}\$ are not simultaneously singular valued decomposable.