spectral theorem for compact self-adjoint operators

紧自伴算子的谱定理

Analysis, Measure & PDE · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a compact self-adjoint operator on a real or complex separable Hilbert space, an orthonormal basis of eigenvectors exists; nonzero eigenvalues are real with finite multiplicity and, if infinitely many, tend to zero when listed with multiplicity. Kernel vectors complete the basis.

实或复可分 Hilbert 空间上的紧自伴算子具有由特征向量组成的标准正交基；非零特征值为实数、重数有限，若按重数列出有无穷多个，则趋于零；算子核中的特征向量补全整个空间的基。

Example

For $(Tx)_n=x_n/(n+1)$ on $\ell^2$ with $n\ge1$, the coordinate vectors form an eigenbasis and the nonzero eigenvalues $1/(n+1)$ tend to zero.

Usage and distinctions

This is the compact self-adjoint eigenbasis theorem, not the general spectral-measure theorem for arbitrary self-adjoint operators. Zero can be a spectral value without being an eigenvalue, and its eigenspace, when present, need not have finite dimension.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Notes on Partial Differential EquationsAppendix 4.B.1, Theorem 4.37, printed p.122: real nonzero eigenvalues, finite-dimensional nonzero eigenspaces and eigenbasis including zero eigenspace · Checked 2026-10-15
2. MIT OpenCourseWare — 18.102 Spring 2021, Complete Lecture NotesLecture 22, Theorem 232 (Spectral theorem), printed pp.117–118: separable Hilbert-space statement and kernel completion · Checked 2026-10-15

Translation sources

1. 许天周、李炳照 — 线性正则变换及其应用，科学出版社预览§1.6, 定理1.6.5（紧自伴算子的谱定理）, printed p.13; used as Chinese naming evidence, with separability stated explicitly in this record · Checked 2026-10-15

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