Order Bounded Operators

Abstract

In the present section we shall prove an important result which lies at the foundations of operator theory in Riesz spaces. Briefly stated, it says that if E and F are Riesz spaces with F Dedekind complete and T : E → F is a linear operator, then T is regular if and only if T is order bounded. In other words, using the notations introduced in section 18, we have L _r (E, F) = L _b (E, F). Moreover (and this is the really interesting point), the vector space L _b (E, F) is now a Dedekind complete Riesz space with the set of positive operators (from E into F) as positive cone. This implies that every T ∈ L _b (E,F) can be written as T = T⁺ - T^-, where T⁺ and T^- are positive operators such that T⁺ =T∨0, T^- = (-T) ∨0 and T⁺∧T^- = 0. The case that F = ℝ is of special interest. Since ℝ is Dedekind complete, it follows that regular linear functionals on E are the same as order bounded linear functionals and the space L _r (E, ℝ) = L _b (E, ℝ) is a Dedekind complete Riesz space. This space, denoted by E~ for convenience, is called the order dual of E. The theorem stating that L _r (E, F) = L _b (E, F) is a Dedekind complete Riesz space is due to L.V. Kantorovitch (1936) in the Soviet Union and to H. Freudenthal (1936) in the Netherlands. The theorem on E~, with further results on order continuous linear functionals (see the next section), is due to F. Riesz (1937) in Hungary who, already in 1928 at the Bologna International Mathematical Congress, presented his ideas about linear functionals on certain ordered vector spaces, which is one of the reasons why lattice ordered vector spaces are now known as Riesz spaces.