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 $T\leq T_1,T_2$?.

New contributor
nowhere is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
  • 4
    $\begingroup$ Welcome to MathOverflow! What if $T_1$ and $T_2$ are orthogonal projections with product $0$? $\endgroup$ – Jochen Glueck May 23 at 7:07
  • 1
    $\begingroup$ Jochen's counterexample works of course for $\dim H =2$ ... $\endgroup$ – Yemon Choi May 23 at 7:27

Your Answer

nowhere is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.