Abstract:
 Let (X，X), (Y，Y) be measurable spaces representing input space and
output space, respectively. For x?X, B?Y we define ν(x，B) as follows.
(1) For any x?X, ν(x，B ) is probability measure on (Y，Y).
(2) For any B?Y, ν(x，B) is measurable function on (X，X).
Then a triple [X，ν，Y ] is called an information channel. Letμ_X be a probability measure on (X，X) representing an input source. We define the following probability measureμ_Y on (Y，Y) as follows.
μ_Y (B) =∫ν(x;B) dμ_X(x)， B?Y,
μ_Y is called an output source. And we define the following probability measureμ_XY on (X ×Y;X × Y) by
μ_XY (C) =∫ν_(x;Cx) dμ_X(x); C?X×Y;
where Cx = {y?Y；(x, y) ?C}.
In this talk we will consider a Gaussian channel, where X; Y are real separable Banach spaces and for each x?X,ν(x; B) is a Gaussian measure on (Y; Y). We state several properties of capacity of Gaussian channel without feedback. And also we state a discrete time Gaussian channel with feedback and give some properties about its capacity.
