Conditional negativity of anti-Loewner matrices

Abstract

Let f be a positive, continuously differentiable function on (0, ∞). We consider matrices K _f of the form We show that for such a function f with f(0) = f ′(0) = 0, all K _f are conditionally negative definite if and only if all are positive semidefinite (p.s.d.), and give the integral representation of f. We also study functions f with K _f  (·, ·) conditionally negative definite or p.s.d.

Keywords:

• anti-Loewner matrix
• Loewner matrix
• conditionally negative definite
• positive semidefinite

AMS Subject Classifications::

• 15A48
• 47A63

Acknowledgements

The authors thank Professor R. Bhatia for the information on 1. The second author is supported by Grant-in-Aid for Scientific Research (C) [KAKENHI] 20540152.