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Aluthge transform

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In mathematics and more precisely in functional analysis, the Aluthge transformation is an operation defined on the set of bounded operators of a Hilbert space. It was introduced by Ariyadasa Aluthge to study p-hyponormal linear operators.

Definition

Let H {\displaystyle H} be a Hilbert space and let B ( H ) {\displaystyle B(H)} be the algebra of linear operators from H {\displaystyle H} to H {\displaystyle H} . By the polar decomposition theorem, there exists a unique partial isometry U {\displaystyle U} such that T = U | T | {\displaystyle T=U|T|} and ker ⁡ ( U ) ⊃ ker ⁡ ( T ) {\displaystyle \ker(U)\supset \ker(T)} , where | T | {\displaystyle |T|} is the square root of the operator T ∗ T {\displaystyle T^{*}T} . If T ∈ B ( H ) {\displaystyle T\in B(H)} and T = U | T | {\displaystyle T=U|T|} is its polar decomposition, the Aluthge transform of T {\displaystyle T} is the operator Δ ( T ) {\displaystyle \Delta (T)} defined as:

Δ ( T ) = | T | 1 2 U | T | 1 2 . {\displaystyle \Delta (T)=|T|^{\frac {1}{2}}U|T|^{\frac {1}{2}}.}

More generally, for any real number λ ∈ [ 0 , 1 ] {\displaystyle \lambda \in } , the λ {\displaystyle \lambda } -Aluthge transformation is defined as

Δ λ ( T ) := | T | λ U | T | 1 − λ ∈ B ( H ) . {\displaystyle \Delta _{\lambda }(T):=|T|^{\lambda }U|T|^{1-\lambda }\in B(H).}

Example

For vectors x , y ∈ H {\displaystyle x,y\in H} , let x ⊗ y {\displaystyle x\otimes y} denote the operator defined as

∀ z ∈ H x ⊗ y ( z ) = ⟨ z , y ⟩ x . {\displaystyle \forall z\in H\quad x\otimes y(z)=\langle z,y\rangle x.}

An elementary calculation shows that if y ≠ 0 {\displaystyle y\neq 0} , then Δ λ ( x ⊗ y ) = Δ ( x ⊗ y ) = ⟨ x , y ⟩ ‖ y ‖ 2 y ⊗ y . {\displaystyle \Delta _{\lambda }(x\otimes y)=\Delta (x\otimes y)={\frac {\langle x,y\rangle }{\lVert y\rVert ^{2}}}y\otimes y.}

Notes

  1. Aluthge, Ariyadasa (1990). "On p-hyponormal operators for 0 < p < 1". Integral Equations Operator Theory. 13 (3): 307–315. doi:10.1007/bf01199886.
  2. Chabbabi, Fadil; Mbekhta, Mostafa (June 2017). "Jordan product maps commuting with the λ-Aluthge transform". Journal of Mathematical Analysis and Applications. 450 (1): 293–313. doi:10.1016/j.jmaa.2017.01.036.

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