Operator monotone function

In linear algebra, operator monotone functions are an important type of real-valued function, fully classified by Charles Löwner in 1934.Template:R They are closely related to operator concave and operator convex functions, and are encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality.Template:RTemplate:R Operator monotone functions are called in other contexts complete Bernstein function, Nevanlinna function, Pick function or class (S) function.Template:R

Definition

A function $f:I\to ℝ$ defined on an interval $I\subseteq ℝ$ 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 $f(A)-f(B)\ge 0$ 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 $B$). The function $f$ is said to be n-matrix monotone (or just n-monotone) if the above holds for any matrices $A$ and $B$ of size $n$ (but not necessarily of other sizes).

Notation

This definition is frequently expressed with the notation that is now defined.

Write $A\ge 0$ to indicate that a matrix $A$ is positive semi-definite and write $A\ge B$ to indicate that the difference $A-B$ of two matrices $A$ and $B$ satisfies $A-B\ge 0$ (that is, $A-B$ is positive semi-definite).

With $f:I\to ℝ$ 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 $f:I\to ℝ$ defined on an interval $I\subseteq ℝ$ 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 $A\ge B$ then $f(A)\ge f(B).$

Löwner’s theorem on holomorphic extension

Löwner’s theoremTemplate:R states that a function $f$ is operator monotone if and only if it allows an analytic continuation to the upper half-plane with non-negative imaginary part.

More generally, a function $f(z)\ge 0$ for $z\ge 0$ is operator monotone if and only if it extends to a holomorphic function on $ℂ\setminus (-\mathrm{\infty },0]$ such that

$\begin{array}{cccc}\mathrm{Im}f(z)& \ge 0& & \text{if }\mathrm{Im}z\ge 0,\\ \mathrm{Im}f(z)& \le 0& & \text{if }\mathrm{Im}z\le 0.\end{array}$

which can be summarized as $\mathrm{Im}f(z)\mathrm{Im}z\ge 0$.

Relation to Bernstein functions

Operator monotone functions are a special type of Bernstein function. If we write the Bernstein representation of the Bernstein function $f$ as$$f(t)=a+bt+{\int }_0^{\mathrm{\infty }}(1-e^{-tx})\mu (dx),$$then $f$ is operator monotone if and only if the measure $\mu$ has a density function and this function is completely monotone, which explains why such a function $f$ is also called a complete Bernstein function.

See also

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Further reading

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