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Let $H$ be a separable Hilbert space, and $K(H)$ the corresponding space of compact operators. Consider the "unit sphere" $S:=\{T\in K(H)|T\geq 0\text{ and }||T||=1\}$. Is it true that, given any pair of operators $T_1,T_2\in S$, there exists another operator $T\in S$ such that $T\leq T_1,T_2$?.

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    $\begingroup$ Welcome to MathOverflow! What if $T_1$ and $T_2$ are orthogonal projections with product $0$? $\endgroup$ – Jochen Glueck May 23 at 7:07
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    $\begingroup$ Jochen's counterexample works of course for $\dim H =2$ ... $\endgroup$ – Yemon Choi May 23 at 7:27

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