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.
defined on an interval
is said to be operator monotone if whenever
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
are the values of the matrix function induced by
(which are matrices of the same size as
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
satisfies
is positive semi-definite).
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
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
Hermitian matrices
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