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.^[1] It is closely allied to the operator concave and operator convex functions, and is encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality.^{[2] [3]}

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

Notation

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

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

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

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

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• Schilling, R.; Song, R.; Vondraček, Z. (2010). Bernstein functions. Theory and Applications. Studies in Mathematics. Vol. 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. arXiv:1112.0098 . doi:10.1016/j.laa.2013年01月02日2. 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年01月01日7.

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