Linear Operators: Spectral theory |
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Page 1179
... PROOF . We saw in the course of proving Theorem 25 that the mapping MK which sends a scalar - valued function with the Fourier transform f ( ) into the vector - valued function whose nth component has the Fourier transform f ...
... PROOF . We saw in the course of proving Theorem 25 that the mapping MK which sends a scalar - valued function with the Fourier transform f ( ) into the vector - valued function whose nth component has the Fourier transform f ...
Page 1724
... PROOF . By the preceding lemma and by Corollary 11 it suffices to show that ( Tf , g ) = ( f , Sg ) for f in D ( T ) and g in D ( S ) . By Green's formula , proved in the last paragraph of Section 2 , this equation is valid if ƒ and g ...
... PROOF . By the preceding lemma and by Corollary 11 it suffices to show that ( Tf , g ) = ( f , Sg ) for f in D ( T ) and g in D ( S ) . By Green's formula , proved in the last paragraph of Section 2 , this equation is valid if ƒ and g ...
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
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