Linear Operators, Part 2 |
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Page 1279
... interval of the real axis . The interval I can be open , half - open , or closed . The interval [ a , ∞ ) is considered to be half - open ; the interval ( −∞ , ∞ ) to be open . Thus a closed interval is a compact set . An end point ...
... interval of the real axis . The interval I can be open , half - open , or closed . The interval [ a , ∞ ) is considered to be half - open ; the interval ( −∞ , ∞ ) to be open . Thus a closed interval is a compact set . An end point ...
Page 1474
... interval . Our second assertion follows immediately from the preceding lemma . Q.E.D. 45 LEMMA . At most one eigenvalue of T lies in any interval J „ · PROOF . Suppose that this is false . Let 21 and 22 be points in Jn ' both being ...
... interval . Our second assertion follows immediately from the preceding lemma . Q.E.D. 45 LEMMA . At most one eigenvalue of T lies in any interval J „ · PROOF . Suppose that this is false . Let 21 and 22 be points in Jn ' both being ...
Page 1605
... interval [ 0 , ∞ ) , then 7 has no boundary values at infinity . ( 2 ) In the interval [ 0 , ∞ ) , suppose that there exists a positive continuously differentiable function M such that ( a ) ( b ) ( c ) p ( t ) 1 / 2M ' ( t ) M ( t ) ...
... interval [ 0 , ∞ ) , then 7 has no boundary values at infinity . ( 2 ) In the interval [ 0 , ∞ ) , suppose that there exists a positive continuously differentiable function M such that ( a ) ( b ) ( c ) p ( t ) 1 / 2M ' ( t ) M ( t ) ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero