Math 2.0

Operator theory Definition

Operator monotone function

Builds on IntervalHermitian matrixLoewner orderMatrix functionPositive semidefinite matrixSpectral theorem for Hermitian matricesMonotone function

Definition

Let I⊆ℝI \subseteq \R be an interval and let f ⁣:I→ℝf \colon I \to \R. Let AA be an n×nn \times n Hermitian matrix with all eigenvalues in II. By the spectral theorem, A=Udiag⁡(λ1,…,λn)U∗A = U \operatorname{diag}(\lambda_1, \dots, \lambda_n) U^* with UU unitary. The matrix function f(A)f(A) is Udiag⁡(f(λ1),…,f(λn))U∗U \operatorname{diag}(f(\lambda_1), \dots, f(\lambda_n)) U^*; it does not depend on the choice of UU. For Hermitian matrices AA and BB of the same size, A≤BA \le B in the Loewner order means that B−AB - A is positive semidefinite. The function ff is operator monotone on II if the following holds for every n≥1n \ge 1. For all n×nn \times n Hermitian matrices AA and BB with eigenvalues in II, if A≤BA \le B, then f(A)≤f(B)f(A) \le f(B).

Remarks

For n=1n = 1 the condition says that ff is nondecreasing, so an operator monotone function is a monotone function. Example: f(t)=−1/tf(t) = -1/t on (0,∞)(0, \infty). If 0<A≤B0 < A \le B, then B−1/2AB−1/2≤IB^{-1/2} A B^{-1/2} \le I, so its eigenvalues lie in (0,1](0, 1], so B1/2A−1B1/2≥IB^{1/2} A^{-1} B^{1/2} \ge I, which gives A−1≥B−1A^{-1} \ge B^{-1}, that is, −A−1≤−B−1-A^{-1} \le -B^{-1}. Non-example: f(t)=t2f(t) = t^2 on [0,∞)[0, \infty). Take A=(1111)A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} and B=(2111)B = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}, both positive semidefinite, with B−A=diag⁡(1,0)≥0B - A = \operatorname{diag}(1, 0) \ge 0. Then B2−A2=(3110)B^2 - A^2 = \begin{pmatrix} 3 & 1 \\ 1 & 0 \end{pmatrix} has determinant −1-1, so it has a negative eigenvalue and A2≤B2A^2 \le B^2 fails. The same definition with self-adjoint operators on a Hilbert space gives the same class of functions.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Hermitian matrix, concept, no evidence levelHermitian matrixconceptIdentity matrix, object, no evidence levelIdentity matrixobjectInterval, concept, no evidence levelIntervalconceptLoewner order, concept, no evidence levelLoewner orderconceptMatrix, definition, no evidence levelMatrixdefinitionMatrix function, concept, no evidence levelMatrix functionconceptMonotone function, concept, no evidence levelMonotone functionconceptPositive semidefinite matrix, concept, no evidence levelPositive semidefinite…conceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionBasis, definition, no evidence levelBasisdefinitionEigenvector, definition, no evidence levelEigenvectordefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionSpectral theorem, statement, no evidence levelSpectral theoremstatementSpectral theorem for Hermitian matrices, statement, no evidence levelSpectral theorem for …statementOperator monotone function, definition, no evidence levelOperator monotone fun…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hermitian matrix concept
  2. Identity matrix object
  3. Interval concept
  4. Loewner order concept
  5. Matrix definition
  6. Matrix function concept
  7. Monotone function concept
  8. Positive semidefinite matrix concept
  9. Vector space definition
  10. Square matrix definition
  11. Transpose definition
  12. Inner product definition
  13. Linear combination definition
  14. Linear map definition
  15. Linear subspace definition
  16. Conjugate transpose definition
  17. Inner product space definition
  18. Orthogonal vectors definition
  19. Linear independence definition
  20. Endomorphism definition
  21. Linear span definition
  22. Hermitian matrix definition
  23. Unitary matrix definition
  24. Orthonormal family definition
  25. Adjoint of a linear map definition
  26. Eigenvalue definition
  27. Spanning set definition
  28. Self-adjoint operator definition
  29. Basis definition
  30. Eigenvector definition
  31. Orthonormal basis definition
  32. Spectral theorem statement
  33. Spectral theorem for Hermitian matrices statement
  34. Operator monotone function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Operator_monotone_function
About this page
TagP2NS0V36TH
URIm2://P2NS0V36TH
Kinddefinition
WritingW0 stub
MSC47A63, 15A45, 26A48
DefinesOperator monotone function
UsesInterval, Hermitian matrix, Loewner order, Matrix function, Positive semidefinite matrix, Spectral theorem for Hermitian matrices, Monotone function
Changed2 November 2026

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