A Fredholm operator is a specific type of bounded linear operator that arises in functional analysis, particularly in the study of integral and differential equations. It is defined on a Hilbert space (or a Banach space) and has certain important characteristics related to its kernel, range, and index. ### Definition: Let \( X \) and \( Y \) be Banach spaces, and let \( T: X \to Y \) be a bounded linear operator.
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A closed operator is Fredholm when it has closed range and finite-dimensional kernel and cokernel. It is upper semi-Fredholm when only closed range and finite kernel are required, and lower semi-Fredholm when only closed range and finite cokernel are required.