|
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space ''H'', the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do not appear in the finite-dimensional case, i.e. for matrices. The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator. In particular, the spectral properties of compact operators resemble those of square matrices. This article first summarizes the corresponding results from the matrix case before discussing the spectral properties of compact operators. The reader will see that most statements transfer verbatim from the matrix case. The spectral theory of compact operators was first developed by F. Riesz. == Spectral theory of matrices == The classical result for square matrices is the Jordan canonical form, which states the following: Theorem. Let ''A'' be an ''n'' × ''n'' complex matrix, i.e. ''A'' a linear operator acting on C''n''. If ''λ''1...''λk'' are the distinct eigenvalues of ''A'', then C''n'' can be decomposed into the invariant subspaces of ''A'' : The subspace ''Yi'' = ''Ker''(''λi'' − ''A'')''m'' where ''Ker''(''λi'' − ''A'')''m'' = ''Ker''(''λi'' − ''A'')''m''+1. Furthermore, the poles of the resolvent function ''ζ'' → (''ζ'' − ''A'')−1 coincide with the set of eigenvalues of ''A''. 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「spectral theory of compact operators」の詳細全文を読む スポンサード リンク
|