Linear Operators, Part 2 |
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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 , F , so that a , F is in H ( C ) . This completes the proof of the direct part of ( i ) of the present lemma . = To prove the converse , let F be in H ( C ) and let 1Ê be in H ) ( C ) . Let us agree to consider that ...
... present proof , F , F , so that a , F is in H ( C ) . This completes the proof of the direct part of ( i ) of the present lemma . = To prove the converse , let F be in H ( C ) and let 1Ê 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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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 domain 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 linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero