Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1629
Nelson Dunford, Jacob T. Schwartz. CHAPTER XIV Linear Partial Differential Equations and Operators 1. Introduction The Cauchy Problem , Local Dependence In this chapter , we shall discuss a variety of theorems having to do with linear ...
Nelson Dunford, Jacob T. Schwartz. CHAPTER XIV Linear Partial Differential Equations and Operators 1. Introduction The Cauchy Problem , Local Dependence In this chapter , we shall discuss a variety of theorems having to do with linear ...
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 ...
Page 1705
... partial differential operator of order at most p - 1 . Τεφ = It follows from ( 1 ) , from Lemmas 3.22 , 3.18 and 3.6 ( iv ) , and on placing ( fo S ) fe , that the distribution f satisfies the partial differential equation ( 2 ) Σaj ...
... partial differential operator of order at most p - 1 . Τεφ = It follows from ( 1 ) , from Lemmas 3.22 , 3.18 and 3.6 ( iv ) , and on placing ( fo S ) fe , that the distribution f satisfies the partial differential equation ( 2 ) Σaj ...
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 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 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