= Compact perturbation invariance of Fredholm operators
{title2=$L\text{ Fredholm}\Longleftrightarrow L+K\text{ Fredholm}$}
For a <Fredholm operator> $L$ on a <Hilbert space>, invert its restriction from the kernel complement to its closed range, extending by zero on the range complement. The resulting bounded $B$ has $BL=I-P$ and $LB=I-Q$ with finite-rank projections. Adding a <compact operator> $K$ gives two-sided inverses modulo compact errors. Such a parametrix forces finite kernel, a lower bound on the kernel complement, closed range and finite adjoint kernel; the latter is the <cokernel>. Repeating with $-K$ proves the equivalence without assuming self-adjointness.
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