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 is an eigenvalue and hence for some non - zero x in § 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 is an eigenvalue and hence for some non - zero x in § we have Tx = λα , and hence , since T TE , we ...
Page 1116
... belongs to the Hilbert - Schmidt class C2 . If we let Ap1 = y / 2p ,, then 4 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 Ap1 = y / 2p ,, then 4 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 ) = 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 ) = 0 on [ 0 , ∞ ) which is not square - integrable but ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero