Linear Operators: Spectral operators |
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Page 1099
... singular matrices , it is sufficient . to consider the case in which T is non - singular . Then A = ( TT * ) 1/2 is also non - singular and if U = A¬1T , UU * = A - 1A2A - 1 I , then U is unitary , and T AU . Let Bo U - 1AP - 1 . Then ...
... singular matrices , it is sufficient . to consider the case in which T is non - singular . Then A = ( TT * ) 1/2 is also non - singular and if U = A¬1T , UU * = A - 1A2A - 1 I , then U is unitary , and T AU . Let Bo U - 1AP - 1 . Then ...
Page 1584
... singular at one or both of the endpoints of the interval of definition . The exploratory work of Hilb [ 2 ] in 1909 ... singular endpoint . It was Hermann Weyl , however , who brought together these loose threads and developed a unified ...
... singular at one or both of the endpoints of the interval of definition . The exploratory work of Hilb [ 2 ] in 1909 ... singular endpoint . It was Hermann Weyl , however , who brought together these loose threads and developed a unified ...
Page 1919
... Singular element in a B - algebra IX.1.2 ( 861 ) Singular element in a ring , ( 40 ) non - singular operator , ( 45 ) Singular set function , definition , III.4.12 ( 131 ) derivatives of , III.12.6 ( 214 ) Lebesgue decomposition theorem ...
... Singular element in a B - algebra IX.1.2 ( 861 ) Singular element in a ring , ( 40 ) non - singular operator , ( 45 ) Singular set function , definition , III.4.12 ( 131 ) derivatives of , III.12.6 ( 214 ) Lebesgue decomposition theorem ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero