Linear Operators: Spectral theory |
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Page 1301
... independent of the set A , ... , An - 1 › and hence has at least n + 1 independent boundary values at a . But this is impossible by Corollary 22. Q.E.D. 24 COROLLARY . ( Weyl - Kodaira ) Let τ be a formally self adjoint formal ...
... independent of the set A , ... , An - 1 › and hence has at least n + 1 independent boundary values at a . But this is impossible by Corollary 22. Q.E.D. 24 COROLLARY . ( Weyl - Kodaira ) Let τ be a formally self adjoint formal ...
Page 1306
... independent solutions of ( T - λ ) σ0 square integrable at a or b when ( 2 ) 0. There are four possibilities as shown by the discussion above . = Number of linearly independent solutions square - integrable : ( i ) ( ii ) ( iii ) ( iv ) ...
... independent solutions of ( T - λ ) σ0 square integrable at a or b when ( 2 ) 0. There are four possibilities as shown by the discussion above . = Number of linearly independent solutions square - integrable : ( i ) ( ii ) ( iii ) ( iv ) ...
Page 1311
... independent boundary conditions defining T is equal to the number of linearly independent solutions of the equation τf belong to L2 ( I ) . - = O which PROOF . Let v be the number of linearly independent solutions of τή = 0 belonging to ...
... independent boundary conditions defining T is equal to the number of linearly independent solutions of the equation τf belong to L2 ( I ) . - = O which PROOF . Let v be the number of linearly independent solutions of τή = 0 belonging to ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero