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 1684
... present lemma for the special case k 1 , m = 0. We shall con- sequently assume for the remainder of the present proof that k = 1 , m = 0 . Our hypothesis then is the fact that every derivative of order 1 of F belongs to L ( E ) , where ...
... present lemma for the special case k 1 , m = 0. We shall con- sequently assume for the remainder of the present proof that k = 1 , m = 0 . Our hypothesis then is the fact that every derivative of order 1 of F belongs to L ( E ) , where ...
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 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