Functional Analysis (8): Spectral Theory — Decomposing Operators

Fri, 15 Oct 2021 09:00:00 +0000

When I first saw the word “spectrum” used for an operator I assumed it was a fancy synonym for “set of eigenvalues.” That is the right intuition for matrices and for compact operators, and it is exactly what one wants in introductory linear algebra. The trouble is that it is wrong as soon as the operator is not compact. The position operator $$(Mf)(x) = x f(x)$$ on $$L^2[0, 1]$$ has no eigenvalues: any eigenfunction would have to satisfy $x f(x) = \lambda f(x)$ a.e., which forces $$f = 0$$ everywhere away from a single point, hence $$f = 0$$ in $$L^2$$ . And yet the operator is clearly not invertible, since $\lambda I - M$ is multiplication by $x - \lambda$ , which fails to be boundedly invertible whenever $\lambda \in [0, 1]$ .

Functional Analysis (6): Bounded Linear Operators and the Big Theorems

Mon, 11 Oct 2021 09:00:00 +0000

Why This Article Is Where the Theory Catches Fire#

For five articles I have been building scaffolding: metric and normed spaces, Hilbert spaces, dual spaces, weak topologies. None of those individually felt very impressive — I am, after all, just doing topology and linear algebra in slightly more general settings than usual. The point at which functional analysis genuinely delivers is right here, in the three great theorems of Banach space operator theory: the Uniform Boundedness Principle, the Open Mapping Theorem, and the Closed Graph Theorem. Each of these takes a piece of “pointwise” or “set-theoretic” data — pointwise boundedness, surjectivity, closedness of the graph — and concludes a global structural property — uniform boundedness, openness, continuity — that has no analog in finite dimensions because finite-dimensional linear algebra makes them all true automatically.

Operators on Chen Kai Blog

Functional Analysis (8): Spectral Theory — Decomposing Operators

Functional Analysis (6): Bounded Linear Operators and the Big Theorems

Why This Article Is Where the Theory Catches Fire#