Is the cone of positive definite operators open?

A positive definite operator is a linear operator T on a Hilbert space such that for all nonzero x, <Tx,x> is positive. The concept of positive definite operators is an extension of that of positive definite matrices.

The set of positive definite operators form a cone, that is, it is closed under positive scaling and addition.

We all know the cone of positive semidefinite matrices is closed and the cone of positive definite matrices is open. But what for the operator case? I know the cone of positive SEMIdefinite operators is closed. But I am uncertain whether the cone of positive definite operators is open or not.
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