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Let $L$ be a linear operator from Banach space $X$ to $Y$. Is the dimension of the kernel of the adjoint of $L$ the same as the dimension of the cokernel? The cokernel is $Y/(Im L)$.

Also, is the index of operators for which this quantity is defined on, same some subspace of the bounded linear operators from $X$ to $Y$, continuous on this set? This is the case for Fredholm operators, but I was wondering if it was true in more general settings.

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    ... the orthogonal projection $P$ onto a subspace of a Hilbert space, so consider $I + tP$. For real $t \neq -1$ this operator is self-adjoint and invertible, so has zero cokernel, while for $t = -1$ you get a cokernel isomorphic to the range of $P$ which can be of arbitrary dimension.2011-11-17

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