Integr. Equ. Oper. Theory (2018) 90:7 https://doi.org/10.1007/s00020-018-2426-x Published online February 28, 2018 c  Springer International Publishing AG, part of Springer Nature 2018 Integral Equations and Operator Theory Preserver Problems Related to Quasi-Arithmetic Means of Invertible Positive Operators Marcell Ga´ al and Gerg˝ o Nagy Abstract. In this paper we mainly discuss different preserver problems on the cone of positive definite matrices which are related to certain quasi-arithmetic means. Mathematics Subject Classification. Primary: 15B48, 47A64, 47B49, Sec- ondary: 53C22. Keywords. Positive definite matrices, Quasi-arithmetic means, Nonlin- ear preservers, Geodesics. 1. Introduction and Statement of the Main Results Means form a fundamental concept in mathematics, originally they are intro- duced for the averaging of real numbers. A mean M : I 2 → I on an interval I is defined as a binary operation satisfying the inequalities min{x, y}≤ M (x, y) ≤ max{x, y} (x, y ∈ I ). Such objects have been intensively studied for a long time by many researchers, their investigation forms a broad field of mathematics. Among them, quasi-arithmetic means are one of the most ba- sic quantities and they are particularly important. For details on such means the interested reader can consult with e.g. the short monograph [2] and the references therein. As for means of other objects, in [9] Kubo and Ando estab- lished the theory of operator means which are certain operations on the cone of positive operators on a Hilbert space. In the finite dimensional case, that notion reduces to means of positive semidefinite matrices which are widely used and investigated in several areas of mathematics. In this paper, we will be mainly concerned with another kind of means, namely the quasi-arithmetic means of positive definite matrices. In order to introduce them, we fix some notation used throughout the paper. Let d ≥ 2 be a given integer. We will denote by M d , H d , H + d and P d the algebra of d × d complex matrices with unit I , the linear space of Hermitian matrices and the cones of positive semidefinite and positive definite matrices, respectively.