# oa.operator algebras – Lower bounds in the space of compact operators

Let $$H$$ be a separable Hilbert space, and $$K(H)$$ the corresponding space of compact operators. Consider the “unit sphere” $$S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$$. Is it true that, given any pair of operators $$T_1,T_2in S$$, there exists another operator $$Tin S$$ such that $$Tleq T_1,T_2$$?.