Kunze Linear Algebra · Problem 6.2.13

Elementary Canonical Forms

Mathematics — Methods · integral-operator-no-characteristic-values

Problem

Let VV be the vector space of all functions from RR into RR which are continuous, i.e., the space of continuous real-valued functions on the real line. Let TT be the linear operator on VV defined by (Tf)(x)=∫0xf(t) dt.(Tf)(x) = \int_0^x f(t)\, dt. Prove that TT has no characteristic values.
Intuitive explanation

The key insight is that characteristic values (eigenvalues) would require a nonzero function ff satisfying ∫0xf(t) dt=λf(x)\int_0^x f(t)\, dt = \lambda f(x) for all xx and some scalar λ\lambda. Differentiate both sides to relate ff to itself, then use boundary conditions to derive a contradiction. This shows that no such eigenfunction can exist in the space of continuous functions, no matter what λ\lambda you try.

Intuitive explanation

The key insight is that characteristic values (eigenvalues) would require a nonzero function ff satisfying ∫0xf(t) dt=λf(x)\int_0^x f(t)\, dt = \lambda f(x) for all xx and some scalar λ\lambda. Differentiate both sides to relate ff to itself, then use boundary conditions to derive a contradiction. This shows that no such eigenfunction can exist in the space of continuous functions, no matter what λ\lambda you try.

Related theory

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