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

For a Hilbert space $H$ , let $\mathcal I_0$ , $\mathcal I_1$ , and $\mathcal I_2$ denote respectively the elementary, trace-class, and Hilbert–Schmidt operator ideals. Shale's subgroups are

GL(H)_i=\{I+X\in GL(H):X\in\mathcal I_i\},\qquad i=0,1,2.

Thus:

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

Remarks

$GL(H)_0$ is dense in $GL(H)_1$ in the trace-class topology and in $GL(H)_2$ in the Hilbert–Schmidt topology.