Let \(\mathbb{M}_d,~\mathbb{H}_d\) and \(\mathbb{P}_d\) denote the sets of complex square matrices, complex Hermitian matrices and positive definite matrices of order \(d\), respectively. A real-valued function \(f\) on \((0,\infty)\) is said to be operator monotone if \(A \geq B\) implies that \(f(A) \geq f(B)\), where \(A\geq B\) stands for \(A-B\) being positive semidefinite with \(A,B \in \mathbb{H}_d\). A real-valued function \(k\) on \((0,\infty)\) is called operator convex if \(k(\lambda A + (1-\lambda)B) \leq \lambda k(A) + (1-\lambda) k(B)\) for all \(\lambda \in (0,1)\) and for all \(A,B \in \mathbb{P}_d\) for any positive integer \(d\). \(k\) is called operator concave if \(-k\) is operator convex. Let \(\mathcal K\) denote the class of functions \(k: (0,\infty) \rightarrow (0,\infty)\) which are operator convex, satisfying the symmetry condition \(xk(x)=k(x^{-1})\) and the normalization condition \(k(1)=1\). The authors prove an integral representation for functions that belong to \(\mathcal K\) using which necessary and sufficient conditions for a function to belong to \(\mathcal K\) are given. These statements are given in terms of operator convexity, operator concavity and operator monotonicity.
A linear map \(\varPhi:\mathbb{M}_d \rightarrow \mathbb{M}_d\) is called positive if it is positivity preserving, namely \(\varPhi (\mathbb {P}_d) \subseteq \bar {\mathbb{P}_d}\), where \(\bar {\mathbb{P}_d}\) denotes the set of complex positive semidefinite matrices. \(\varPhi\) is called completely positive if \(\varPhi \otimes I\) is positive on \(\mathbb{M}_d \otimes \mathbb{M}_n\) for all positive integers \(n\). Given \(k \in \mathcal K\), consider the function \(\phi^k(x,y):=(1/y)k(x/y)\) for \(x,y >0\). Let \(D\) be unitarily diagonalizable so that there is a unitary matrix \(U\) such that \(D=U \operatorname{diag}(\lambda_1,\dots, \lambda_d) U^*\). Define \(\varOmega^k_D(X)=U([\phi(\lambda_i,\lambda_j)] \circ [U^*XU])U^*\), where \(\circ\) denotes the Hadamard entrywise product. The authors also study the problem of when these operators are completely positive. A complete analysis of the behaviour of one-parameter families for which either the map or its inverse is completely positive is presented.