Operator monotone function

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In linear algebra, the operator monotone function is an important type of real-valued function, fully classified by Charles Löwner in 1934. 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.

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Definition

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} ).

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).}

References

Löwner, K.T. (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift. 38: 177–216. doi:10.1007/BF01170633. S2CID 121439134. http://eudml.org/doc/168495 ↩
"Löwner–Heinz inequality". Encyclopedia of Mathematics. https://www.encyclopediaofmath.org/index.php/Löwner–Heinz_inequality ↩
Chansangiam, Pattrawut (2013). "Operator Monotone Functions: Characterizations and Integral Representations". arXiv:1305.2471 [math.FA]. /wiki/ArXiv_(identifier) ↩

Definition

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References