A function ${\displaystyle f:I\to \mathbb {R} }$  defined on an interval ${\displaystyle I\subseteq \mathbb {R} }$  is said to be operator monotone if whenever ${\displaystyle A}$  and ${\displaystyle B}$  are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of ${\displaystyle f}$  and whose difference ${\displaystyle A-B}$  is a positive semi-definite matrix, then necessarily ${\displaystyle f(A)-f(B)\geq 0}$  where ${\displaystyle f(A)}$  and ${\displaystyle f(B)}$  are the values of the matrix function induced by ${\displaystyle f}$  (which are matrices of the same size as ${\displaystyle A}$  and ${\displaystyle B}$ ).

Notation

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

With ${\displaystyle f:I\to \mathbb {R} }$  and ${\displaystyle A}$  as in the theorem's statement, the value of the matrix function ${\displaystyle f(A)}$  is the matrix (of the same size as ${\displaystyle A}$ ) defined in terms of its ${\displaystyle A}$ 's spectral decomposition ${\displaystyle A=\sum _{j}\lambda _{j}P_{j}}$  by

The definition of an operator monotone function may now be restated as:

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

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• 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.