Linear Operators: Spectral theory |
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Page 1142
... present theorem in the range 1 < p ≤ 2 now follows at once from its validity in the range 2 ≤ p ≤ ∞ and from Lemma 9.14 . Q.E.D. In what follows , we will use the symbols p and ʼn to denote the continuous extension to the classes C ...
... present theorem in the range 1 < p ≤ 2 now follows at once from its validity in the range 2 ≤ p ≤ ∞ and from Lemma 9.14 . Q.E.D. In what follows , we will use the symbols p and ʼn to denote the continuous extension to the classes C ...
Page 1675
... present proof , F = √1F , so that a1F is in H ( C ) . This completes the proof of the direct part of ( i ) of the present lemma . П To prove the converse , let F be in H ( C ) and let a1Ê be in H ( C ) . Let us agree to consider that ...
... present proof , F = √1F , so that a1F is in H ( C ) . This completes the proof of the direct part of ( i ) of the present lemma . П To prove the converse , let F be in H ( C ) and let a1Ê be in H ( C ) . Let us agree to consider that ...
Page 1703
... present lemma . Q.E.D. 6. The Elliptic Boundary Value Problem Can the boundary value theory and the spectral theory of Chapter XIII be generalized to partial differential operators ? In the present section it will be seen that it can ...
... present lemma . Q.E.D. 6. The Elliptic Boundary Value Problem Can the boundary value theory and the spectral theory of Chapter XIII be generalized to partial differential operators ? In the present section it will be seen that it can ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero