Operator monotone function

In linear algebra, the operator monotone function is an important type of real-valued function, first described by Charles Löwner in 1934. It is closely allied to the operator concave and operator concave functions, and is encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality.

Definition

See also: Trace inequality § Operator monotone

A function $f:I\to \mathbb {R}$ defined on an interval $I\subseteq \mathbb {R}$ is said to be operator monotone if for all $n,$ and all $n\times n$ Hermitian matrices $A,B$ with eigenvalues in $I,$ the following holds, $$A\geq B\Rightarrow f(A)\geq f(B),$$ where the inequality $A\geq B$ means that the operator $A-B\geq 0$ is positive semi-definite.

Further reading

• Schilling, R.; Song, R.; Vondraček, Z. (2010). Bernstein functions. Theory and Applications. Studies in Mathematics. Vol. 37. de Gruyter, Berlin. doi:10.1515/9783110215311. ISBN 9783110215311.
• Hansen, Frank (2013). "The fast track to Löwner's theorem". Linear Algebra and Its Applications. 438 (11): 4557–4571. arXiv:1112.0098. doi:10.1016/j.laa.2013.01.022. S2CID 119607318.
• Chansangiam, Pattrawut (2015). "A Survey on Operator Monotonicity, Operator Convexity, and Operator Means". International Journal of Analysis. 2015: 1–8. doi:10.1155/2015/649839.

See also

• Matrix function
• Trace inequality