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.{{r|low}} 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.{{r|eom}}{{r|pat}}

Definition

A function $f : I \to \Reals$ defined on an interval $I \subseteq \Reals$ 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 $f(A) - f(B) \geq 0$

where $f(A)$ and $f(B)$ are the values of the matrix function induced by $f$ (which are matrices of the same size as $A$ and $B$ ).

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).

With $f : I \to \Reals$ and $A$ as in the theorem's statement, the value of the matrix function $f(A)$ is the matrix (of the same size as $A$ ) defined in terms of its $A$ 's spectral decomposition $A = \sum_j \lambda_j P_j$ by

$f(A) = \sum_j f(\lambda_j)P_j ~,$

where the $\lambda_j$ are the eigenvalues of $A$ with corresponding projectors $P_j.$

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

A function $f : I \to \Reals$ defined on an interval $I \subseteq \Reals$ 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 $f(A) \geq f(B).$

References

{{reflist|refs=

{{cite journal|author = Löwner, K.T.|journal = Mathematische Zeitschrift|pages = 177–216|title = Über monotone Matrixfunktionen|url = http://eudml.org/doc/168495|volume = 38|year = 1934|doi = 10.1007/BF01170633|s2cid = 121439134|url-access = subscription}}

{{cite arXiv|eprint = 1305.2471|last1 = Chansangiam|first1 = Pattrawut|title = Operator Monotone Functions: Characterizations and Integral Representations|class = math.FA|year = 2013}}

{{cite web|title=Löwner–Heinz inequality|website=Encyclopedia of Mathematics|url= https://www.encyclopediaofmath.org/index.php/Löwner–Heinz_inequality}}

}}

Operator monotone function

Definition

See also

References

Further reading