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 n 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 n to denote the continuous extension to the classes C ...
Page 1679
... present lemma it suffices to show that ( 4 ) G ( S ̧ ¥ ( • ) q ( • − y ) f ( y ) dy ) = √ , G ( v ( · ) p ( • —y ) ) f ( y ) dy , where GF . Let K2 be a compact subset of I containing in its interior a second compact set outside of ...
... present lemma it suffices to show that ( 4 ) G ( S ̧ ¥ ( • ) q ( • − y ) f ( y ) dy ) = √ , G ( v ( · ) p ( • —y ) ) f ( y ) dy , where GF . Let K2 be a compact subset of I containing in its interior a second compact set outside of ...
Page 1692
... present lemma . Q.E.D. 1 9 COROLLARY . The conclusions of Corollary 6 and Lemma 8 remain valid even if the open set I of these results is replaced by the cube C = { x = E " | | x ; | < π , j = 1 , ... , n } . PROOF . It was observed ...
... present lemma . Q.E.D. 1 9 COROLLARY . The conclusions of Corollary 6 and Lemma 8 remain valid even if the open set I of these results is replaced by the cube C = { x = E " | | x ; | < π , j = 1 , ... , n } . PROOF . It was observed ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero