-The infinite-dimensional QL factorization is a subtly thing: its defined when the operator has non-positive Fredholm index, and if the Fredholm index is not zero, it may not be unique. For the Bull head matrix `A`, here are plots of `ql(A-λ*I).L[1,1]` alongside the image of the symbol `A`, which depicts the essential spectrum of `A` and where the Fredholm index changes. Note we have two plots as the regions with negative Fredholm index have multiple QL factorizations. Where the Fredholm index is positive, the QL factorization doesn't exist and is depected in black.
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