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.^[2][3] Operator monotone functions are called in other contexts complete Bernstein function, Nevanlinna function, Pick function or class (S) function.^[4]

A function f : I → R {\displaystyle f:I\to \mathbb {R} } defined on an interval I ⊆ R {\displaystyle I\subseteq \mathbb {R} } is said to be operator monotone if whenever A {\displaystyle A} and B {\displaystyle B} are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of f {\displaystyle f} and whose difference A − B {\displaystyle A-B} is a positive semi-definite matrix, then necessarily f ( A ) − f ( B ) ≥ 0 {\displaystyle f(A)-f(B)\geq 0} where f ( A ) {\displaystyle f(A)} and f ( B ) {\displaystyle f(B)} are the values of the matrix function induced by f {\displaystyle f} (which are matrices of the same size as A {\displaystyle A} and B {\displaystyle B} ). The function f {\displaystyle f} is said to be n-matrix monotone (or just n-monotone) if the above holds for any matrices A {\displaystyle A} and B {\displaystyle B} of size n {\displaystyle n} (but not necessarily of other sizes).

Notation

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

Write A ≥ 0 {\displaystyle A\geq 0} to indicate that a matrix A {\displaystyle A} is positive semi-definite and write A ≥ B {\displaystyle A\geq B} to indicate that the difference A − B {\displaystyle A-B} of two matrices A {\displaystyle A} and B {\displaystyle B} satisfies A − B ≥ 0 {\displaystyle A-B\geq 0} (that is, A − B {\displaystyle A-B} is positive semi-definite).

With f : I → R {\displaystyle f:I\to \mathbb {R} } and A {\displaystyle A} as in the theorem's statement, the value of the matrix function f ( A ) {\displaystyle f(A)} is the matrix (of the same size as A {\displaystyle A} ) defined in terms of its A {\displaystyle A} 's spectral decomposition A = ∑ j λ j P j {\displaystyle A=\sum _{j}\lambda _{j}P_{j}} by f ( A ) = ∑ j f ( λ j ) P j   , {\displaystyle f(A)=\sum _{j}f(\lambda _{j})P_{j}~,} where the λ j {\displaystyle \lambda _{j}} are the eigenvalues of A {\displaystyle A} with corresponding projectors P j . {\displaystyle P_{j}.}

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

A function f : I → R {\displaystyle f:I\to \mathbb {R} } defined on an interval I ⊆ R {\displaystyle I\subseteq \mathbb {R} } said to be operator monotone if (and only if) for all positive integers n , {\displaystyle n,} and all n × n {\displaystyle n\times n} Hermitian matrices A {\displaystyle A} and B {\displaystyle B} with eigenvalues in I , {\displaystyle I,} if A ≥ B {\displaystyle A\geq B} then f ( A ) ≥ f ( B ) . {\displaystyle f(A)\geq f(B).}

Löwner’s theorem^[1] states that a function f {\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 f ( z ) ≥ 0 {\displaystyle f(z)\geq 0} for z ≥ 0 {\displaystyle z\geq 0} is operator monotone if and only if it extends to a holomorphic function on C ∖ ( − ∞ , 0 ] {\displaystyle \mathbb {C} \setminus (-\infty ,0]} such that

Im ⁡ f ( z ) ≥ 0 if  Im ⁡ z ≥ 0 , Im ⁡ f ( z ) ≤ 0 if  Im ⁡ z ≤ 0. {\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 Im ⁡ f ( z ) Im ⁡ z ≥ 0 {\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 f {\displaystyle f} as f ( t ) = a + b t + ∫ 0 ∞ ( 1 − e − t x ) μ ( d x ) , {\displaystyle f(t)=a+bt+\int _{0}^{\infty }\left(1-e^{-tx}\right)\mu (dx),} then f {\displaystyle f} is operator monotone if and only if the measure μ {\displaystyle \mu } has a density function and this function is completely monotone, which explains why such a function f {\displaystyle f} is also called a complete Bernstein function.

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• 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.
• Boţ, Radu Ioan; Csetnek, Ernö Robert; Hendrich, Christopher (1 April 2015). "Inertial Douglas–Rachford splitting for monotone inclusion problems". Applied Mathematics and Computation. 256: 472–487. doi:10.1016/j.amc.2015.01.017.