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 whenever $A$ and $B$ are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of $f$ and whose difference $A-B$ is a positive semi-definite matrix, then necessarily $f(A)-f(B)\geq 0.$

Notation

This definition is frequently expressed with the notation that is now defined. Write $A\geq 0$ to indicate that a matrix $A$ is positive semi-definite and write $A\geq B$ to indicate that the difference $A-B$ of two matrices $A$ and $B$ satisfies $A-B\geq 0$ (that is, $A-B$ is positive semi-definite). The definition may be restated as follows.

A function $f:I\to \mathbb {R}$ defined on an interval $I\subseteq \mathbb {R}$ said to be operator monotone if (and only if) for all positive integers $n,$ and all $n\times n$ Hermitian matrices $A$ and $B$ with eigenvalues in $I,$ if $A\geq B$ then $f(A)\geq f(B).$

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