/sci/ - Science & Math

>>11887609
I'm 24 and I got my masters already.

>>11770069
The answer to your example is NO.
Choose an orthonormal basis [math](e_i)_{i \in I}[/math]. I will denote by [math]v \otimes w[/math] the unique operator [math]A \in B(\mathcal{H})[/math] given by the formula [math]Ax = \langle x, w \rangle v[/math].
Then, let
[math]A_k = (-1 + \frac{1}{k})e_1 \otimes e_1 - e_k \otimes e_k[/math],
[math]S = \{A_k : k \geq 1 \}[/math],
Then [math]T = -e_1 \otimes e_1[/math] does not belong to [math]S+P[/math], but it belongs to the closure.