Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1027
... belongs to the spectrum of both T and ET . Suppose that λ 0 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that 2 is an eigenvalue and hence for some non - zero x in § we have Tx λα , and hence , since T TE ...
... belongs to the spectrum of both T and ET . Suppose that λ 0 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that 2 is an eigenvalue and hence for some non - zero x in § we have Tx λα , and hence , since T TE ...
Page 1116
... belongs to the Hilbert - Schmidt class C2 . If we let Aq , y / 2q , then A is plainly self adjoint and A belongs -p / 2 = to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 − p / 2 ) −1 . Thus , by Lemma 9 , TBA belongs to ...
... belongs to the Hilbert - Schmidt class C2 . If we let Aq , y / 2q , then A is plainly self adjoint and A belongs -p / 2 = to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 − p / 2 ) −1 . Thus , by Lemma 9 , TBA belongs to ...
Page 1602
... belongs to the essential spectrum of 7 ( Hartman and Wintner [ 14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let ƒ be a real solution of the equation ( λ — 7 ) ƒ = 0 on [ 0 , ∞ ) which is not square - integrable ...
... belongs to the essential spectrum of 7 ( Hartman and Wintner [ 14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let ƒ be a real solution of the equation ( λ — 7 ) ƒ = 0 on [ 0 , ∞ ) which is not square - integrable ...
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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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero