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.