nominal
发表于 2025-3-26 23:14:14
(Square) Roots of Bounded OperatorsLet . ∈ .(.). We say that . ∈ .(.) is a square root of . if .. = ..
Tdd526
发表于 2025-3-27 01:59:52
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blight
发表于 2025-3-27 06:39:48
SpectrumLet . ∈ .(.) where . is a complex Hilbert space. The set . is called the spectrum of ..
联合
发表于 2025-3-27 10:26:30
Spectral Radius, Numerical RangeLet . be in .(.). The spectral radius of . is defined as
菊花
发表于 2025-3-27 14:09:24
Functional CalculiThe functional calculus aims to define .(.) where . is a fixed operator, and . belongs to some classes of functions defined in domains containing .(.), say. We already know that this is possible for any polynomial .. We also know how to define the exponential of . at an undergraduate level (this will be recalled in Chap. .).
EXUDE
发表于 2025-3-27 21:46:12
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calamity
发表于 2025-3-27 23:17:32
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aesthetic
发表于 2025-3-28 05:56:46
Similarity and Unitary EquivalenceClearly, . and . have the same eigenvalues which, in this setting, means that . and . have equal spectra. To see why . and . are not unitarily equivalent, remember that two unitarily equivalent operators are simultaneously (e.g.) self-adjoint. Since . is self-adjoint and . is not, it follows that they cannot be unitarily equivalent.
Proponent
发表于 2025-3-28 08:06:15
The Sylvester EquationConsider the operator equation: . where ., ., . ∈ .(.) are given and . ∈ .(.) is the unknown. This equation is more commonly known as the Sylvester equation.
consolidate
发表于 2025-3-28 11:29:58
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