Operator monotone function

In linear algebra, operator monotone functions are an important type of real-valued function, fully classified by Charles Löwner in 1934.¹ 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.²³ Operator monotone functions are called in other contexts complete Bernstein function, Nevanlinna function, Pick function or class (S) function.⁴

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 ${\displaystyle B}$). The function ${\displaystyle f}$ is said to be n-matrix monotone (or just n-monotone) if the above holds for any matrices ${\displaystyle A}$ and ${\displaystyle B}$ of size ${\displaystyle n}$ (but not necessarily of other sizes).

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 $${\displaystyle f(A)=\sum _{j}f(\lambda _{j})P_{j}~,}$$ 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 ${\displaystyle f(A)\geq f(B).}$

Löwner’s theorem¹ states that a function ${\displaystyle f}$ 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 ${\displaystyle f(z)\geq 0}$ for ${\displaystyle z\geq 0}$ is operator monotone if and only if it extends to a holomorphic function on ${\displaystyle \mathbb {C} \setminus (-\infty ,0]}$ such that

${\displaystyle {\begin{aligned}\operatorname {Im} f(z)&\geq 0\qquad &&{\text{if }}\operatorname {Im} z\geq 0,\\\operatorname {Im} f(z)&\leq 0\qquad &&{\text{if }}\operatorname {Im} z\leq 0.\end{aligned}}}$

which can be summarized as ${\displaystyle \operatorname {Im} f(z)\operatorname {Im} z\geq 0}$.

Operator monotone functions are a special type of Bernstein function. If we write the Bernstein representation of the Bernstein function ${\displaystyle f}$ as$${\displaystyle f(t)=a+bt+\int _{0}^{\infty }\left(1-e^{-tx}\right)\mu (dx),}$$then ${\displaystyle f}$ is operator monotone if and only if the measure ${\displaystyle \mu }$ has a density function and this function is completely monotone, which explains why such a function ${\displaystyle f}$ is also called a complete Bernstein function.