The convex-concave characteristics of Gaussian channel capacity functions

In this correspondence, we give several inherent properties of the capacity function of a Gaussian channel with and without feedback by using some operator inequalities and matrix analysis. We give a new proof method which is different from the method appearing in: K. Yanagi and H. W. Chen, "Operator inequality and its application to information theory," Taiwanese J. Math., vol. 4, no. 3, pp. 407-416, Sep. 2000. We obtain the following results: C_n,Z(P) and C_n,FB,Z(P) are both concave functions of P, C_n,Z(P) is a convex function of the noise covariance matrix and C_n,FB,Z(P) is a convex-like function of the noise covariance matrix. This new proof method is very elementary and the results shall help study the capacity of Gaussian channel. Finally, we state a conjecture concerning the convexity of C_n,FB,·(P).