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In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. In contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach. For example, the spectral theory of compact operators on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal. A corresponding result holds for normal compact operators on Hilbert spaces. (More generally, the compactness assumption can be dropped. But, as stated above, the techniques used are less routine.) This article will discuss a few results for compact operators on Hilbert space, starting with general properties before considering subclasses of compact operators. == Some general properties == Let ''H'' be a Hilbert space, ''L''(''H'') be the bounded operators on ''H''. ''T'' ∈ ''L''(''H'') is a compact operator if the image of each bounded set under ''T'' is relatively compact. We list some general properties of compact operators. If ''X'' and ''Y'' are Hilbert spaces (in fact ''X'' Banach and ''Y'' normed will suffice), then ''T:'' ''X'' → ''Y'' is compact if and only if it is continuous when viewed as a map from ''X'' with the weak topology to ''Y'' (with the norm topology). (See , and note in this reference that the uniform boundedness will apply in the situation where ''F'' ⊆ ''X'' satisfies (∀φ ∈ Hom(''X'', ''K'')) sup < ∞, where ''K'' is the underlying field. The uniform boundedness principle applies since Hom(''X'', ''K'') with the norm topology will be a Banach space, and the maps ''x * *'':Hom(''X'',''K'') → ''K'' are continuous homomorphisms with respect to this topology.) The family of compact operators is a norm-closed, two-sided, *-ideal in ''L''(''H''). Consequently, a compact operator ''T'' cannot have a bounded inverse if ''H'' is infinite-dimensional. If ''ST'' = ''TS'' = ''I'', then the identity operator would be compact, a contradiction. If a sequence of bounded operators ''Sn'' → ''S'' in the strong operator topology and ''T'' is compact, then ''SnT'' converges to ''ST'' in norm. For example, consider the Hilbert space ''l''2(N), with standard basis . Let ''Pm'' be the orthogonal projection on the linear span of . The sequence converges to the identity operator ''I'' strongly but not uniformly. Define ''T'' by ''Ten'' = (1/''n'')2 · ''en''. ''T'' is compact, and, as claimed above, ''PmT'' → ''I T'' = ''T'' in the uniform operator topology: for all ''x'', : Notice each ''Pm'' is a finite-rank operator. Similar reasoning shows that if ''T'' is compact, then ''T'' is the uniform limit of some sequence of finite-rank operators. By the norm-closedness of the ideal of compact operators, the converse is also true. The quotient C *-algebra of ''L''(''H'') modulo the compact operators is called the Calkin algebra, in which one can consider properties of an operator up to compact perturbation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Compact operator on Hilbert space」の詳細全文を読む スポンサード リンク
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