Polar Decomposition and Partial Isometries

Polar bear and Maine Coon cat contemplating a partial isometry 
𝑈
:
𝐻
→
𝐻
′
U:H→H 
′
  on the snow.
Trying to understand partial isometries: a polar bear, a Maine Coon, and a rainbow arrow from H\mathcal{H} to H′\mathcal{H}’.

In this note I revisit the polar decomposition T=U∣T∣ for bounded (possibly antilinear) operators between Hilbert spaces, emphasizing the role of partial isometries and their initial and final subspaces.

The diagram in the snow shows exactly what our polar bear and cat are contemplating: a partial isometry whose initial subspace sits inside H, and whose final subspace lives inside H‘.

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