Linear Operators: Spectral theory |
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Page 1305
... boundary values at a and b respectively ) , then B ( f ) = 0 is said to be a mixed boundary condition . A set of boundary conditions is said to be separated if it ( or , more generally , some set of boundary conditions equivalent to it ) ...
... boundary values at a and b respectively ) , then B ( f ) = 0 is said to be a mixed boundary condition . A set of boundary conditions is said to be separated if it ( or , more generally , some set of boundary conditions equivalent to it ) ...
Page 1307
... boundary values at a and D1 , D2 are boundary values at b , such that ( Tf , g ) - ( f , Tg ) = C1 ( f ) C2 ( g ) —C2 ( f ) C1 ( g ) + D1 ( f ) D2 ( g ) —D2 ( f ) D1 ( g ) , f , ge D ( T1 ( T ) ) . PROOF . Let A be any boundary value ...
... boundary values at a and D1 , D2 are boundary values at b , such that ( Tf , g ) - ( f , Tg ) = C1 ( f ) C2 ( g ) —C2 ( f ) C1 ( g ) + D1 ( f ) D2 ( g ) —D2 ( f ) D1 ( g ) , f , ge D ( T1 ( T ) ) . PROOF . Let A be any boundary value ...
Page 1310
... boundary conditions at a , and exactly one solution y ( t , λ ) of ( r − λ ) y = 0 square - integrable at b and satisfying the boundary conditions at b . - PROOF . We shall show the theorem is true in each of the four cases discussed ...
... boundary conditions at a , and exactly one solution y ( t , λ ) of ( r − λ ) y = 0 square - integrable at b and satisfying the boundary conditions at b . - PROOF . We shall show the theorem is true in each of the four cases discussed ...
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