Linear Operators: Spectral theory |
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Page 906
... real numbers as a subclass of the class of all complex numbers . In particular , every T e B ( § ) can be written uniquely in the form T A + iB , where A and B are Hermitian operators . Clearly , A and B must be given by the formulae ...
... real numbers as a subclass of the class of all complex numbers . In particular , every T e B ( § ) can be written uniquely in the form T A + iB , where A and B are Hermitian operators . Clearly , A and B must be given by the formulae ...
Page 1251
... real numbers . A necessary and sufficient condition that there exist a non- negative measure μ defined on the Borel sets of the real line such that √∞ \ t \ " μ ( dt ) < ∞ and is that ∞ mn = ( dt ) , n = 0 , 1 , 2 , ... , n ∞- i ...
... real numbers . A necessary and sufficient condition that there exist a non- negative measure μ defined on the Borel sets of the real line such that √∞ \ t \ " μ ( dt ) < ∞ and is that ∞ mn = ( dt ) , n = 0 , 1 , 2 , ... , n ∞- i ...
Page 1744
... real numbers without a finite limit point ) is bounded below . This , however , follows imme- diately from Corollary ... real axis and the countable set of real numbers o ( V ) . Then , ( i ) there exists a real number K such that | R ...
... real numbers without a finite limit point ) is bounded below . This , however , follows imme- diately from Corollary ... real axis and the countable set of real numbers o ( V ) . Then , ( i ) there exists a real number K such that | R ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero