Linear Operators: Spectral operators |
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Page 1629
... differential operators . Since the theory of linear partial differential operators is vast and highly ramified , we shall only touch upon a number of its aspects , with the intention of displaying a bouquet of applications of functional ...
... differential operators . Since the theory of linear partial differential operators is vast and highly ramified , we shall only touch upon a number of its aspects , with the intention of displaying a bouquet of applications of functional ...
Page 1633
... derivatives not initially required of them . A third category of formal partial differential operators is the parabolic , typified by the operator a 02 δαι dx2 This sort of operator is closely related to the theory of semi - groups ...
... derivatives not initially required of them . A third category of formal partial differential operators is the parabolic , typified by the operator a 02 δαι dx2 This sort of operator is closely related to the theory of semi - groups ...
Page 1703
... derivatives , proving the 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 ...
... derivatives , proving the 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 ...
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 Haar measure 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 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 unique unitary vanishes vector zero