Lucas N. F. Teles

Lucas N. F. Teles

Undergrad in Applied Mathematics, Universidade de São Paulo

The spectrum of linear operators

1 minute read

Published:

Essentially, spectral theory seeks to further the study of eigenvalues for the case of linear operators $T:X \to X$ on infinite dimensional spaces $X$.

For that purpose, while the standard definition of an eigenvalue still works (that is, $\lambda \in \mathbb{C}$ such that $\text{Ker}(T-\lambda I) \neq {0}$), it needs to be complemented to take care of “infinite dimensional artifacts” that these perturbations $\lambda I$ can employ which could cause troubles when inverting or handling the inversion of $T-\lambda$.

We call the set of these “disturbing perturbations” $\lambda$ the spectrum of 𝑇 and it consists of 3 disjoint parts:

  • The discrete spectrum, where $\lambda$ are such that $T-\lambda I$ “can’t be reversed”, that is, $\text{Ker}(T-\lambda I) \neq {0}$ (which accounts for the eigenvalues).
  • The continuous spectrum, where $(T-\lambda I)$ is invertible on a dense subset of the domain, but it’s inverse is unbounded.
  • The residual spectrum, which accounts for perturbations $\lambda$ whose inversion is bounded but isn’t valid for a “sufficiently large” domain ($\overline{\text{Dom}(R_\lambda)} \neq X$).

While the discrete spectrum inherits the geometric intuition of it’s finite dimensional counterpart, the reasoning behind the two remaining spectra isn’t immediate. To that end I share one interesting way I saw (in a StackExchange thread linked bellow) to interpret the continuous spectrum:

If the resolvent operator $R_\lambda = (T-\lambda I)^{-1}$ is unbounded, we have a sequence $(x_n)_{n\in \mathbb{N}}$ in the unit sphere whose image under $R_\lambda$ diverges to infinity. As $R_\lambda$ represents a way to reverse the transformation $T - \lambda$, we may use it to translate the sequence $(x_n)_{n \in \mathbb{N}}$ to $\text{Dom}(T)$, and normalizing the vectors, construct a sequence

\[\tilde{x}\_n = \frac{R_\lambda(T)x_n}{\|R_\lambda(T)x_n\|}, \quad \forall n \in \mathbb{N}\]

in the unit ball of $X$ such that $(T-\lambda)x_n \to 0$.

That is, the continuous spectrum represents the set of scalars whose corresponding “perturbed transformation” have a sequence on the unit sphere along which it’s behavior approximates that of an eigenvector for $\lambda$.

Credit: https://math.stackexchange.com/questions/2087926/meaning-of-the-continuous-spectrum-and-the-residual-spectrum.

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