Knowlpedia

Restricted General Linear Group rGL(H)

Invertible operators whose positive part differs from I by a Hilbert–Schmidt operator

Shale's restricted general linear group is

rGL(H)={T∈GL(H):∣T∣=(T∗T)1/2∈GL(H)2}.rGL(H)=\{T\in GL(H): |T|=(T^*T)^{1/2}\in GL(H)_2\}.
Equivalent characterizations

Equivalently, ∣T∣−I|T|-I is .

Remarks

This group is exactly where the Gaussian measure in §3 is quasi-invariant.

Key properties:

  • Stable under T=u(T)∣T∣T=u(T)|T|.
  • Carries the "change-of-variables" unitary rep on L2(M,n)L_2(M,n).
Examples
  • In finite dimensions, rGL(H)=GL(H)rGL(H)=GL(H).