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Compact operator on Hilbert space
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<p>In the mathematical discipline of <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, <a href="/facts/Compact_operator/KNkcK4mE">compact operators</a> are precisely the closure of <a href="/facts/Finite-rank_operator/jA2kCmCA">finite-rank operators</a> (representable by finite-dimensional matrices) in the <a href="/facts/Topology/2EqW47cU">topology</a> induced by the <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a>. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach. </p><p>For example, the <a href="/facts/Spectral_theory_of_compact_operators/jfNuc5U4">spectral theory of compact operators</a> on <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> takes a form that is very similar to the <a href="/facts/Jordan_canonical_form/M7ezYbnl">Jordan canonical form</a> of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is <a href="/facts/Normal_operator/lwtz7tuF">normal</a>. A corresponding result holds for normal compact operators on Hilbert spaces. More generally, the compactness assumption can be dropped. As stated above, the techniques used to prove results, e.g., the <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a>, in the non-compact case are typically different, involving operator-valued <a href="/facts/Measure_(mathematics)/yCq7zlI4">measures</a> on the <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a>. </p><p>Some results for compact operators on Hilbert space will be discussed, starting with general properties before considering subclasses of compact operators. </p>