Title:Doubly Infinite Matrices: Algebra Needs Help From Analysis
Abstract:A typical step in matrix algebra is elimination, and its
description as a triangular factorization. For a doubly
infinite banded Toeplitz matrix A, that step is made easy by
factoring the polynomial a(z) whose coefficients come from
the diagonals of A. What to do if A is not Toeplitz?
A nice case is a permutation matrix (on Z). Which is the main
diagonal? For the (Toeplitz) example of a shift matrix, the
main diagonal contains the 1's. We identify the correct diagonal
for every banded permutation. Then we consider banded matrices
(not Toeplitz!) as operators on L2(Z) and ask about their factorization.
The help coming from analysis is the theory of Fredholm operators.
