Timeline for Associativity of polar decomposition
Current License: CC BY-SA 2.5
3 events
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| Dec 2, 2010 at 0:18 | comment | added | anon | Ack! I even got the term wrong (my "dilation" is really an extension, and as you point out, things with kernels can't extend to things that don't have them). One can dilate (in the proper sense of the term) any contraction to a unitary, but the ``$A'$'' you get will have to have some junk in the lower left corner. Perhaps it is possible to salvage this idea by understanding what happens with the junk when you do a polar decomposition. But maybe it is no simpler than trying another approach. (I leave my whole answer unedited so that these comments make sense to people who read them.) | |
| Dec 1, 2010 at 20:11 | comment | added | Chris Heunen | The first half of your argument in fact shows the desired result to hold when A is an isometry (even when considering functions between different spaces). But unfortunately, in the second half, not every partial isometry can be dilated to an isometry. For example, $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ is a partial isometry on $\mathbb{C}^2$, and no matter what larger space we embed $\mathbb{C}^2$ in, the kernel will stay nontrivial. | |
| Dec 1, 2010 at 10:28 | history | answered | anon | CC BY-SA 2.5 |