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polar decomposition (Definition)

The polar decomposition of an operator is a generalization of the familiar factorization of a complex number $ z$ in a radial part $ \vert z\vert$ and an angular part $ z/\vert z\vert$.

Let $ \mathscr{H}$ be a Hilbert space, $ x$ a bounded operator on $ \mathscr{H}$. Then there exist a pair $ (h,u)$, with $ h$ a bounded positive operator and $ u$ a partial isometry on $ \mathscr{H}$, such that

$\displaystyle x=uh.$

If we impose the further conditions that $ 1-u^*u$ is the projection to the kernel of $ x$, and $ \ker(h)=\ker(x)$, then $ (h,u)$ is unique, and is called the polar decomposition of $ x$. The operator $ h$ will be $ \vert x\vert$, the square root of $ x^*x$, and $ u$ will be the partial isometry, determined by

  • $ u\xi=0$ for $ \xi \in \ker(x)$
  • $ u(\vert x\vert\xi)=x\xi$ for $ \xi\in \mathscr{H}$.

If $ x$ is a closed, densely defined unbounded operator on $ \mathscr{H}$, the polar decomposition $ (u,h)$ still exists, where now $ h$ will be the unbounded positive operator $ \vert x\vert$ with the same domain $ \mathscr{D}(x)$ as $ x$, and $ u$ still the partial isometry determined by

  • $ u\xi=0$ for $ \xi \in \ker(x)$
  • $ u(\vert x\vert\xi)=x\xi$ for $ \xi\in \mathscr{D}(x)$.

If $ x$ is affiliated with a von Neumann algebra $ M$, both $ u$ and $ h$ will be affiliated with $ M$.



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polar decomposition in von Neumann algebras (Result) by asteroid
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Cross-references: von Neumann algebra, domain, unbounded, unbounded operator, densely defined, closed, square root, kernel, projection, partial isometry, positive operator, bounded, bounded operator, Hilbert space, radial, complex number, operator
There are 4 references to this entry.

This is version 7 of polar decomposition, born on 2006-06-23, modified 2007-07-22.
Object id is 8075, canonical name is PolarDecomposition.
Accessed 2305 times total.

Classification:
AMS MSC: 47A05 (Operator theory :: General theory of linear operators :: General )
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