/sci/ - Science & Math

>>15087799
It's not true in general.
Define [math]T: l^2 \to l^2[/math], [math]T(e_n) = e_{n + 1}[/math] if n is odd and [math]T(e_n) = e_{n - 1}[/math] if [math]n[/math] is even. The operator is clearly unitary, hence [math]TT^* - I = 0[/math]. But [math]T - I[/math] obviously has infinite Hilbert-Schmidt norm, since the sum has an infinite number of equal [math]\| (T - I)e_1 \|^2 = \| e_2 - e_1 \|^2[/math] factors.