Operator monotone function
In linear algebra, operator monotone functions are an important type of real-valued function, fully classified by Charles Löwner in 1934. They are closely related to operator concave and operator convex functions, and are encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality. Operator monotone functions are called in other contexts complete Bernstein function, Nevanlinna function, Pick function or class function.
Definition
A function defined on an interval is said to be operator monotone if whenever and are Hermitian matrices 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 . The function is said to be n-matrix monotone if the above holds for any matrices and of size .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 .
With and as in the theorem's statement, the value of the matrix function is the matrix 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 for all positive integers and all Hermitian matrices and with eigenvalues in if then
Löwner’s theorem on holomorphic extension
Löwner’s theorem states that a function is operator monotone if and only if it allows an analytic continuation to the upper half-plane with non-negative imaginary part.More generally, a function for is operator monotone if and only if it extends to a holomorphic function on such that
which can be summarized as.