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Inequalities for some Monotone Matrix Functions

Published online by Cambridge University Press:  20 November 2018

Marvin Marcus  and
Paul J. Nikolai
Marvin Marcus
Affiliation:
The University of California, Santa Barbara, California
Paul J. Nikolai
Affiliation:
The University of California, Santa Barbara, California
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Let V denote a unitary vector space with inner product (x, y). A self-adjoint linear map T: V → V is positive (positive definite) if (Tx, x) ≧ 0 ((Tx, x) ≧ 0) for all x ≠ 0 in V. We write S ≧ T(S > T) if S and T are self-adjoint and S – T ≧ 0 (S – T > 0). If U is a unitary vector space, a function f: Hom(V, V) → Hom(U, U) is monotone idf S ≧ T implies that f(S) ≧ f(T). If both U and V are taken to be the n-dimensional unitary space Cn of n-tuples of complex numbers with standard inner product, then f is a monotone matrix junction, a notion introduced for a more restrictive class of functions by Löwner (3) which has important applications in pure and applied mathematics. For orientation we refer the reader to (1), where several interesting examples of monotone and related functions are displayed in detail.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 485 - 494
Copyright
Copyright © Canadian Mathematical Society 1969

References

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