Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1218
... measurable functions having the required property is a linear manifold . Furthermore , the characteristic function % of a Borel set e is in this manifold . This follows from the regularity of u , for there is an open set o containing e ...
... measurable functions having the required property is a linear manifold . Furthermore , the characteristic function % of a Borel set e is in this manifold . This follows from the regularity of u , for there is an open set o containing e ...
Page 1221
... measurable sets , and it follows that am is μ - measurable , completing the proof of statement ( i ) . To complete the proof of the theorem , suppose that the functions W1 ( ` , λ ) , . . . , W2 ( , λ ) are not linearly independent for u ...
... measurable sets , and it follows that am is μ - measurable , completing the proof of statement ( i ) . To complete the proof of the theorem , suppose that the functions W1 ( ` , λ ) , . . . , W2 ( , λ ) are not linearly independent for u ...
Page 1341
... measure μ . If { m ;; } is the matrix of densities of μ , with respect to μ , then there exist non- negative u - measurable functions q ;, i = 1 , ... , n , u - integrable over each bounded interval , and u - measurable functions a ...
... measure μ . If { m ;; } is the matrix of densities of μ , with respect to μ , then there exist non- negative u - measurable functions q ;, i = 1 , ... , n , u - integrable over each bounded interval , and u - measurable functions a ...
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