Operator monotone function

In linear algebra, the operator monotone function is an important type of real-valued function, fully classified by Charles Löwner in 1934.^[1] 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.^[2][3]

Definition

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

where the ${\displaystyle \lambda _{j}}$ are the eigenvalues of ${\displaystyle A}$ with corresponding projectors ${\displaystyle P_{j}.}$

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 ${\displaystyle f(A)\geq f(B).}$

See also

• Matrix function – Function that maps matrices to matricesPages displaying short descriptions of redirect targets
• Trace inequality – inequalities involving linear operators on Hilbert spacesPages displaying wikidata descriptions as a fallback

References

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.