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

Definition

edit

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

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

References

edit
  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

edit