Defining the domain of positive real numbers

There's a misunderstanding here. The third "dom" argument is not simply a set over which we solve the equation. There are only a few choices that can be used for the domain argument, and they have very specific effects on how Solve works. An example from the documentation:

If dom is Reals, or a subset such as Integers or Rationals, then all constants and function values are also restricted to be real.

So you can't use e.g.

Solve[x^2 == 1, x, Interval[{0, Infinity}]]

The proper way to do this, as @belisarius said, is to append the constraint to the system of equations:

Solve[x^2 == 1 && x > 0, x]

In version 10 we can also do

Solve[x^2 == 1 && x ∈ Interval[{0, Infinity}], x]

or even

Solve[x^2 == 1, x ∈ Interval[{0, Infinity}]]

New in Mathematica 12 is PositiveReals (and others like NonNegativeIntegers, etc):

Solve[x^2 == 1, x, PositiveReals]

{{x -> 1}}

{Solve [ x^2 == 1, {x}], Solve [ x^2 == 1 && x > 0, {x}]}
(* {{{x -> -1}, {x -> 1}}, {{x -> 1}}}*)