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      "label": "> [!theorem] <a class=\"object-permalink\" href=\"#theorem-13-19\">定理 13.19</a> · 有界厄米算符的谱定理 (Spectral theorem for bounded Hermitian operators)",
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        "quote": "\n\n> [!theorem] <a class=\"object-permalink\" href=\"#theorem-13-19\">定理 13.19</a> · 有界厄米算符的谱定理 (Spectral theorem for bounded Hermitian operators)\n>\n> 设 $A$ 是希尔伯特空间 $\\mathcal{H}$ 上的厄米算符，谱为 $\\Sigma(A)$。根据<a href=\"#theorem-13-17\" class=\"cross-reference\" id=\"xref-05-spectral-theory-10\" data-reference=\"theorem-13-17\">定理 13.17</a>，$\\Sigma(A)$是 $\\mathbb{R}$ 的有界闭子集。存在递增的投影算符族 $E_{\\lambda}(\\lambda \\in \\mathbb{R})$，其中对于 $\\lambda \\leq \\lambda^{\\prime}$有 $P_{\\lambda} \\leq E_{\\lambda^{\\prime}}$ ，这样一来\n>\n> $$\n> E_{\\lambda}=0 \\quad \\text {当} \\lambda<\\inf (\\Sigma(A))\\text{时}, \\quad E_{\\lambda}=I \\quad \\text {当} \\lambda>\\sup (\\Sigma(A))\\text{时}\n> $$\n>\n> 有\n>\n> $$\n> A=\\int_{-\\infty}^{\\infty} \\lambda \\mathrm{d} E_{\\lambda} .\n> $$\n",
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