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. 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

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

Notation

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

With [math]\displaystyle{ f : I \to \Reals }[/math] and [math]\displaystyle{ A }[/math] as in the theorem's statement, the value of the matrix function [math]\displaystyle{ f(A) }[/math] is the matrix (of the same size as [math]\displaystyle{ A }[/math]) defined in terms of its [math]\displaystyle{ A }[/math]'s spectral decomposition [math]\displaystyle{ A = \sum_j \lambda_j P_j }[/math] by [math]\displaystyle{ f(A) = \sum_j f(\lambda_j)P_j ~, }[/math] where the [math]\displaystyle{ \lambda_j }[/math] are the eigenvalues of [math]\displaystyle{ A }[/math] with corresponding projectors [math]\displaystyle{ P_j. }[/math]

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

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

See also

• Matrix function – Function that maps matrices to matrices
• Trace inequality

Further reading

• Schilling, R.; Song, R.; Vondraček, Z. (2010). Bernstein functions. Theory and Applications. Studies in Mathematics. 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. doi:10.1016/j.laa.2013.01.022. 
• 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.