Complex Operators

Abstract

If T is a (linear) operator mapping the real vector space V into the real vector space W, then T has a unique extension as a (linear) operator from V + iV into W + iW by defining T (f + ig) = T f + iTg for all f and g in V. The set L (V, W) of all (linear) operators mapping V into W can be considered thus as a real-linear subspace of the vector space of all complex (linear) operators from V + iV into W + iW. Recall here that a subset K of a complex vector space G is said to be a real-linear subspace of G whenever α f + ßg ∈ K for all f and g in K and all real numbers α,ß. Any T ∈ L(V, W) which is thus extended is called a real (linear) operator from V + iV into W + iW. Any arbitrary operator T ∈ L (V + iV, W + iW) has a unique decomposition T = T₁+iT₂withT₁ and T₂ real. To see this, decompose T f for any f ∈ V into its real and imaginary parts: T f = T₁ f + iT₂ f. Obviously, the thus defined T₁ and T₂ are (linear) operators from V into W which can be extended from V +iV into W + iW. It follows then that Th = T₁ h + iT­₂ h for any h = f + ig ∈ V + iV, and so T = T₁ + iT₂ holds on V + iV with T₁ and T₂ real. This shows that the space L (V + iV, W + iW) is the complexification of L (V, W).