Knowlpedia

Polar Decomposition

Writing T as a unitary/orthogonal part times a positive part

Any invertible bounded operator TT on a Hilbert space has a polar decomposition

T=u(T) ∣T∣,∣T∣=(T∗T)1/2≥0,T=u(T)\,|T|,\quad |T|=(T^*T)^{1/2}\ge 0,

where u(T)u(T) is unitary (complex case) or orthogonal (real case).

Remarks

Key properties (paper use):

  • Used to define via ∣T∣∈GL(H)2|T|\in GL(H)_2.
  • Lets one reduce many proofs to the positive cone (e.g. Sp(K)+Sp(K)^+).
Examples
  • If TT is positive selfadjoint, then u(T)=Iu(T)=I and ∣T∣=T|T|=T.