Projects

(Q34+44) If an operator S can be written as S = TT, then we abbreviate S^1/2 := T. In the mapping sequence x → Tx → TTx, clearly S^1/2 “lies between” S⁰ and S¹. We know that (AB)⁻¹ = B⁻¹A⁻¹, so far existant, and so S⁻¹ = T⁻¹T⁻¹. So we can abbreviate S^−1/2 := T⁻¹, and (S^1/2)⁻¹ = S^−1/2. It is now clear that S^−1/2SS^−1/2 = T⁻¹TTT⁻¹ = Id.

It remains to show that S^−1/2 is a limit of polynomials in S⁻¹ in the strong operator topology.