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Operator monotone function

From Wikipedia, the free encyclopedia

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 concave functions, and is encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality.[2][3]

Definition

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A function defined on an interval is said to be operator monotone if whenever and are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of and whose difference is a positive semi-definite matrix, then necessarily where and are the values of the matrix function induced by (which are matrices of the same size as and ).

Notation

This definition is frequently expressed with the notation that is now defined. Write to indicate that a matrix is positive semi-definite and write to indicate that the difference of two matrices and satisfies (that is, is positive semi-definite).

With and as in the theorem's statement, the value of the matrix function is the matrix (of the same size as ) defined in terms of its 's spectral decomposition by where the are the eigenvalues of with corresponding projectors

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

A function defined on an interval said to be operator monotone if (and only if) for all positive integers and all Hermitian matrices and with eigenvalues in if then

See also

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  • Matrix function – Function that maps matrices to matrices
  • Trace inequality – inequalities involving linear operators on Hilbert spaces

References

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  1. ^ Löwner, K.T. (1934). "Über monotone Matrixfunktionen". Mathematische Zeitschrift. 38: 177–216. doi:10.1007/BF01170633. S2CID 121439134.
  2. ^ "Löwner–Heinz inequality". Encyclopedia of Mathematics.
  3. ^ Chansangiam, Pattrawut (2013). "Operator Monotone Functions: Characterizations and Integral Representations". arXiv:1305.2471 [math.FA].

Further reading

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