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PhyslibAlpha.ProbabilisticTheory.HilbertSpace.TraceClass.GeneralProduct

Products of Hilbert–Schmidt operators

i. Overview

A product of two Hilbert–Schmidt operators is trace class. Every trace-class operator `T` factors as `(U √|T|) √|T|` with both factors Hilbert–Schmidt, so trace-class operators form a two-sided ideal closed under sums and adjoints.

ii. Key results

- `isTraceClass_mul_of_isHilbertSchmidt` : a product of two Hilbert–Schmidt operators is trace class. - `isTraceClass_add` : trace-class operators are closed under addition. - `isTraceClass_mul_mul` : trace-class operators form a two-sided ideal. - `isTraceClass_star` : trace-class operators are closed under the adjoint.

6 declarations

theorem

⟨bi,W∗RSbi⟩=⟨R∗Wbi,Sbi⟩\langle b_i, W^* R S b_i \rangle = \langle R^* W b_i, S b_i \rangle

Let HH be a complex Hilbert space and let R,S,WR, S, W be continuous linear operators on HH. Given a Hilbert basis {bi}i∈w\{b_i\}_{i \in w} for HH, the following identity holds for every index i∈wi \in w: ⟨bi,W∗(RS)bi⟩=⟨(R∗W)bi,Sbi⟩\langle b_i, W^* (R S) b_i \rangle = \langle (R^* W) b_i, S b_i \rangle where W∗W^* and R∗R^* denote the adjoints of WW and RR, respectively.

theorem

The Product of Two Hilbert–Schmidt Operators is Trace Class

Let HH be a complex Hilbert space, and let RR and SS be bounded linear operators on HH. If RR and SS are Hilbert–Schmidt operators, then their product RSR S is a trace-class operator.

theorem

Hilbert–Schmidt Factorization of Trace-Class Operators as T=(U∣T∣)∣T∣T = (U\sqrt{|T|})\sqrt{|T|}

Let HH be a complex Hilbert space and TT be a trace-class operator on HH. Let UU be the polar factor of TT and ∣T∣=T∗T|T| = \sqrt{T^* T} be its absolute value. Then the operators U∣T∣U\sqrt{|T|} and ∣T∣\sqrt{|T|} are both Hilbert–Schmidt operators, and they satisfy the factorization T=(U∣T∣)∣T∣T = (U\sqrt{|T|})\sqrt{|T|}.

theorem

The sum of trace-class operators is trace-class

Let HH be a complex Hilbert space. If TT and T′T' are trace-class operators on HH, then their sum T+T′T + T' is also a trace-class operator.

theorem

ATBA T B is trace-class if TT is trace-class

Let HH be a complex Hilbert space. If TT is a trace-class operator on HH, then for any bounded linear operators AA and BB on HH, the composition ATBA T B is also a trace-class operator. (Recall that a bounded linear operator TT is trace-class if ∑i∈I⟨ei,∣T∣ei⟩<∞\sum_{i \in I} \langle e_i, |T| e_i \rangle < \infty for some Hilbert basis {ei}i∈I\{e_i\}_{i \in I} of HH).

theorem

The adjoint of a trace-class operator is trace-class

Let HH be a complex Hilbert space and T:H→HT: H \to H be a bounded linear operator. If TT is trace-class, then its adjoint T∗T^* is also trace-class.