Problem understanding Wigner's 1939 paper

[facenian]

if subspace A is invariant with respect to a linear operator, is it true that A is invariant with respect to the inverse operator? if not I think threre is a mistake in Wigners paper in "B. Some immediate simplifications" page 13

 

[facenian]

I'm sorry it's my mistake, I didn't read well. At the beginning of the paragraph he says that the subspace is invariant under all lorentz transformation then it must be invariant for $D(L^{-1})$ too

 

[A. Neumaier]

if subspace A is invariant with respect to a linear operator, is it true that A is invariant with respect to the inverse operator?

Yes, if the linear operator is invertible and maps A onto A.

 

[Samy_A]

Yes, if the linear operator is invertible (and hence maps A onto A).

If invariant is defined as $T(A) \subseteq A$, then I think this isn't necessarily true in an infinite dimensional space.

 

[A. Neumaier]

If invariant is defined as $T(A) \subseteq A$, then I think this isn't necessarily true in an infinite dimensional space.

Indeed. In infinite dimensions you need to assume more. I corrected my statement accordingly.