Linear Operators: Spectral theory |
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Page 1086
... Moreover , the series n - 1 D ( s , t ; 2 ) = ∞ ( -2 ) " Σ D ( s , t ) n = 2 ( n − 1 ) ! converges for all 2 , for uxu - almost all [ s , t ] , and D ( s , t ; 2 ) is the kernel which represents the operator D ( 2 ) —2d ( 2 ) I of the ...
... Moreover , the series n - 1 D ( s , t ; 2 ) = ∞ ( -2 ) " Σ D ( s , t ) n = 2 ( n − 1 ) ! converges for all 2 , for uxu - almost all [ s , t ] , and D ( s , t ; 2 ) is the kernel which represents the operator D ( 2 ) —2d ( 2 ) I of the ...
Page 1298
... Moreover , by the continuity of the map g → fig , B1 is a continuous linear functional on D ( T ) . A similar argument holds for B. Thus B1 and B2 are boundary values for t . If g ( t ) = 0 in a neighborhood of a , then fige D ( T ) by ...
... Moreover , by the continuity of the map g → fig , B1 is a continuous linear functional on D ( T ) . A similar argument holds for B. Thus B1 and B2 are boundary values for t . If g ( t ) = 0 in a neighborhood of a , then fige D ( T ) by ...
Page 1480
... Moreover , there is a unique ( up to constant multiple ) eigenfunction în of T as- sociated with μn , and yn has precisely n − 1 zeros . PROOF . Since σ ( T ) ( − ∞ , 20 ) is void , then every point in o ( T ) ( -∞ , ) is isolated ...
... Moreover , there is a unique ( up to constant multiple ) eigenfunction în of T as- sociated with μn , and yn has precisely n − 1 zeros . PROOF . Since σ ( T ) ( − ∞ , 20 ) is void , then every point in o ( T ) ( -∞ , ) is isolated ...
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 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 open set 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 tr(T unique unitary vanishes vector zero