On infinite divisibility of positive definite functions arising from operator means

For a (point-wisely non-negative) positive definite function a certain criterion for its infinite divisibility (i.e., all its fractional powers are also positive definite) is obtained. This criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means. In particular, we determine when the functioncosh(νx)+s′coshx+s(ν∈[0,1];s,s′∈(−1,1]) is infinitely divisible.