Linear Operators: Spectral theory |
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Page 955
... linear functional h , and since , by IX.2.3 , every such function is continuous , it follows that a multiplicative linear functional on A is entirely determined by its restriction to A. Thus , there is a unique point P in M such that ...
... linear functional h , and since , by IX.2.3 , every such function is continuous , it follows that a multiplicative linear functional on A is entirely determined by its restriction to A. Thus , there is a unique point P in M such that ...
Page 1293
... function g in H ( I ) such that t * g and the assertion reduces to = w , So f ( t ) x * g ( t ) dt = 0 which is the hypothesis of the theorem . ( E ) Suppose now that some linear functional q on L2 ( I ) , represented by a function h ...
... function g in H ( I ) such that t * g and the assertion reduces to = w , So f ( t ) x * g ( t ) dt = 0 which is the hypothesis of the theorem . ( E ) Suppose now that some linear functional q on L2 ( I ) , represented by a function h ...
Page 1303
... linear functional on the Hilbert space ( T1 ( t ) ) . If lim → B1 ( f ) B ( f ) exists for each fin D ( T1 ( t ) ) , then , by Theorem II.1.17 , B is a continuous linear func- tional on D ( T1 ( t ) ) . Clearly B ( ƒ ) = 0 for those ƒ ...
... linear functional on the Hilbert space ( T1 ( t ) ) . If lim → B1 ( f ) B ( f ) exists for each fin D ( T1 ( t ) ) , then , by Theorem II.1.17 , B is a continuous linear func- tional on D ( T1 ( t ) ) . Clearly B ( ƒ ) = 0 for those ƒ ...
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