Linear Operators: Spectral theory |
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Page 1027
... belongs to the spectrum of both T and ET . Suppose that λ0 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that λ is an eigenvalue and hence for some non - zero x in H we have Tx λα , and hence , since T TE , we ...
... belongs to the spectrum of both T and ET . Suppose that λ0 belongs to the spectrum of T. Since T is compact , Theorem VII.4.5 shows that λ is an eigenvalue and hence for some non - zero x in H we have Tx λα , and hence , since T TE , we ...
Page 1116
... belongs to the Hilbert - Schmidt class C2- If we let Ap = y / 2p ,, then A is plainly self adjoint and A belongs to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 - p / 2 ) -1 . Thus , by Lemma 9 , T = BA belongs to the ...
... belongs to the Hilbert - Schmidt class C2- If we let Ap = y / 2p ,, then A is plainly self adjoint and A belongs to the class C ,, where r ( 1 - p / 2 ) = p , i.e. , r = p ( 1 - p / 2 ) -1 . Thus , by Lemma 9 , T = BA belongs to the ...
Page 1602
... belongs to the essential spectrum of 7 ( Hartman and Wintner [ 14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let f be a real solution of the equation ( 2-7 ) ƒ = 0 on [ 0 , ∞ ) which is not square - integrable but ...
... belongs to the essential spectrum of 7 ( Hartman and Wintner [ 14 ] ) . ( 48 ) Suppose that the function q is bounded below , and let f be a real solution of the equation ( 2-7 ) ƒ = 0 on [ 0 , ∞ ) which is not square - integrable but ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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59 other sections not shown
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