Functional Analysis (7): Compact Operators — The Bridge to Finite Dimensions

Wed, 13 Oct 2021 09:00:00 +0000

I owe my fondness for compact operators to a small embarrassment. As an undergraduate I assumed that infinite-dimensional linear algebra would feel exotic everywhere. It does not. There is a wide and well-mapped suburb of operator theory in which everything one learned about symmetric matrices – eigenvalues, orthogonal eigenvectors, the spectral decomposition – comes back almost unchanged, just with eigenvalues tailing off to zero instead of a finite list. That suburb is the world of compact operators, and the price of admission is a single condition: the operator must squeeze the unit ball into a relatively compact set. Once that condition is met, nearly everything follows: the spectrum is countable, nonzero eigenvalues are isolated with finite-dimensional eigenspaces, the Fredholm alternative holds, and integral equations of the second kind become as tractable as linear systems. The line between matrices and infinite-dimensional operators ceases to be a wall and becomes a permeable membrane.

Compact-Operators on Chen Kai Blog

Functional Analysis (7): Compact Operators — The Bridge to Finite Dimensions