Shale's Subgroups GL(H)₀, GL(H)₁, GL(H)₂

For a Hilbert space $H$, Shale defines subgroups of $GL(H)$ by $GL(H)_i=\{I+X:\ X\in \mathcal I_i\}$, where:

• $i=0$: $X$ elementary (finite-rank/finite-dimensional support).
• $i=1$: $X$ trace-class .
• $i=2$: $X$ Hilbert–Schmidt .

Key property: $GL(H)_0$ is dense in $GL(H)_1$, dense in $GL(H)_2$ (in the corresponding norms).