Endomorphisms of Inner Product Spaces

Abstract

Let V be an inner product space. If α is a self-adjoint endomorphism of V and v ∈ V then (math) and so (α(v),v) ∈ ℝ. An endomorphism a of V is positive if and only if it is self-adjoint and satisfies the condition that 0 (α(v),v) ∈ ℝ for all 0 _V ≠ v ∈ V. Thus, for example, σ₁ ∈ End(V) is positive. We note immediately that if α ∈ End(V) is positive then α must be a monomorphism since α(v) = 0 _v implies that (α(v),v) = 0 and so we must have v = 0 _v .