Linear Operators: Spectral theory |
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Page 961
... PROOF . If we write y for y ( e ) , then ( † , v ) = √ „ f ( x ) μ ( x ) dx = √ „ † ( x − 0 ) ч ( x ) dx R = √ „ ƒ ( 0 — x ) y ( x ) dx = ( ƒ * * y ) ( 0 ) = d ( ƒ * y ) . Since the operation T ( f ) of convolution by ƒ commutes ...
... PROOF . If we write y for y ( e ) , then ( † , v ) = √ „ f ( x ) μ ( x ) dx = √ „ † ( x − 0 ) ч ( x ) dx R = √ „ ƒ ( 0 — x ) y ( x ) dx = ( ƒ * * y ) ( 0 ) = d ( ƒ * y ) . Since the operation T ( f ) of convolution by ƒ commutes ...
Page 1459
... PROOF . We use the notations of the proof of Theorem 8. By Lemma 29 and Theorem 28 it is sufficient to show that t ' is finite below in order to conclude that is finite below 2. But it was shown in the proof of Theorem 8 that c may be ...
... PROOF . We use the notations of the proof of Theorem 8. By Lemma 29 and Theorem 28 it is sufficient to show that t ' is finite below in order to conclude that is finite below 2. But it was shown in the proof of Theorem 8 that c may be ...
Page 1750
... proof of Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2. The theorem is false if no ... PROOF ( of Theorem 1 ) . The proof will be given in a series of steps , some of which will be proofs of auxiliary ...
... proof of Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2. The theorem is false if no ... PROOF ( of Theorem 1 ) . The proof will be given in a series of steps , some of which will be proofs of auxiliary ...
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