Square-roots of operators

Stephan Paukner :: syslog

Friday, February 9. 2007

Square-roots of operators

Master's Thesis

(Q34+44) If an operator S can be written as S = TT, then we abbreviate S1/2 := T. In the mapping sequence x → Tx → TTx, clearly S1/2 “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 (S1/2)−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.

Posted by Stephan Paukner in Master's Thesis at 12:41 | Comments (0) | Trackbacks (0)
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