Self-commutator approximants
Copyright policy )Self-commutator approximants
This paper deals with minimizing ‖ B − ( X ∗ X − X X ∗ ) ‖ p \| B - (X^* X - X X^*) \|_p , where B B is fixed, self-adjoint and B ∈ C p B \in \mathcal {C}_p , and where X X varies such that B X = X B BX = XB and X ∗ X − X X ∗ ∈ C p X^* X - X X^* \in \mathcal {C}_p , 1 ≤ p > ∞ 1 \leq p > \infty . (Here, C p \mathcal {C}_p , 1 ≤ p > ∞ 1 \leq p > \infty , denotes the von Neumann-Schatten class and ‖ ⋅ ‖ p \| \cdot \|_p its norm.) The upshot of this paper is that ‖ B − ( X ∗ X − X X ∗ ) ‖ p \| B - (X^* X - X X^*) \|_p , 1 ≤ p > ∞ 1 \leq p > \infty , is minimized if, and for 1 > p > ∞ 1 > p > \infty only if, X ∗ X − X X ∗ = 0 X^* X - X X^* = 0 , and that the map X → ‖ B − ( X ∗ X − X X ∗ ) ‖ p p X \rightarrow \| B - (X^* X - X X^*) \|_p^p , 1 > p > ∞ 1 > p > \infty , has a critical point at X = V X = V if and only if V ∗ V − V V ∗ = 0 V^* V - V V^* = 0 (with related results for normal B B if p = 1 p = 1 or 2 2 ).
- Middlesex University United Kingdom
Linear operator approximation theory, Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.), self-adjointness, approximation by commutators: Heisenberg uncertainty principle, normality, Commutators, derivations, elementary operators, etc., self-commutators, Norms (inequalities, more than one norm, etc.) of linear operators, Schatten--von Neumann class, Subnormal operators, hyponormal operators, etc.
Linear operator approximation theory, Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.), self-adjointness, approximation by commutators: Heisenberg uncertainty principle, normality, Commutators, derivations, elementary operators, etc., self-commutators, Norms (inequalities, more than one norm, etc.) of linear operators, Schatten--von Neumann class, Subnormal operators, hyponormal operators, etc.
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