Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 85
Page 1221
... continuous as functions on c to G ( S , Σ , v ) . Let λ be fixed in c . Then , by Lemma 18 , there is an m such that the restrictions W1 ( ` , 2。) , . . . , Ŵ2 - 1 ( ` , 2 % ) of W1 ( ` , 26 ) , ... , Wp - 1 ( ' , 2 ) to Sm are linearly ...
... continuous as functions on c to G ( S , Σ , v ) . Let λ be fixed in c . Then , by Lemma 18 , there is an m such that the restrictions W1 ( ` , 2。) , . . . , Ŵ2 - 1 ( ` , 2 % ) of W1 ( ` , 26 ) , ... , Wp - 1 ( ' , 2 ) to Sm are linearly ...
Page 1303
... continuous linear functional on the Hilbert space D ( T1 ( 7 ) ) . If lim , → B¿ ( ƒ ) B ( f ) exists for each fin D ( T , ( T ) ) , then , by Theorem II.1.17 , B is a continuous linear func- tional on D ( T , ( t ) ) . Clearly B ( f ) ...
... continuous linear functional on the Hilbert space D ( T1 ( 7 ) ) . If lim , → B¿ ( ƒ ) B ( f ) exists for each fin D ( T , ( T ) ) , then , by Theorem II.1.17 , B is a continuous linear func- tional on D ( T , ( t ) ) . Clearly B ( f ) ...
Page 1679
... function y vanishes ; we may evidently and shall henceforth suppose that K2 is the closure of its interior 2. Then , for sufficiently large m , G may be extended from Co ( 2 ) to a continuous linear functional on Cm ( K ) . Indeed , if ...
... function y vanishes ; we may evidently and shall henceforth suppose that K2 is the closure of its interior 2. Then , for sufficiently large m , G may be extended from Co ( 2 ) to a continuous linear functional on Cm ( K ) . Indeed , if ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
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