Linear Operators, Part 2 |
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Page 1120
... assume for simplicity of statement that Hilbert space is separable . Subdiagonal representations of an operator are connected with the study of its invariant subspaces . Thus , the key to the situation that we wish to analyze is the ...
... assume for simplicity of statement that Hilbert space is separable . Subdiagonal representations of an operator are connected with the study of its invariant subspaces . Thus , the key to the situation that we wish to analyze is the ...
Page 1429
... Assume that ( a ) ( b ) - | a , ( t ) is bounded away from zero ; ax ( t ) is bounded , 0 ≤ k ≤ n . Then has no boundary values at infinity . PROOF . Without loss of generality , assume that a = 0. Let t ' be the formally self adjoint ...
... Assume that ( a ) ( b ) - | a , ( t ) is bounded away from zero ; ax ( t ) is bounded , 0 ≤ k ≤ n . Then has no boundary values at infinity . PROOF . Without loss of generality , assume that a = 0. Let t ' be the formally self adjoint ...
Page 1629
... assumed to be defined for [ t , ... , t ] in D , to be symmetric in the indices i , 2 , . . . , i ,, and , unless the ... assume for 1629 Linear Partial Differential Equations and Operators Introduction The Cauchy Problem, Local Dependence.
... assumed to be defined for [ t , ... , t ] in D , to be symmetric in the indices i , 2 , . . . , i ,, and , unless the ... assume for 1629 Linear Partial Differential Equations and Operators Introduction The Cauchy Problem, Local Dependence.
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