Linear Operators: Spectral theory |
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Page 1305
... boundary condition either at a or at b ( so that , by Theorem 19 , the equation B ( f ) = 0 may be written as B1 ( ƒ ) ... conditions is said to be separated if it ( or , more generally , some set of boundary conditions equivalent to it ) ...
... boundary condition either at a or at b ( so that , by Theorem 19 , the equation B ( f ) = 0 may be written as B1 ( ƒ ) ... conditions is said to be separated if it ( or , more generally , some set of boundary conditions equivalent to it ) ...
Page 1310
... conditions . Let 20. Then the boundary conditions are real , and there is exactly one solution ( t , λ ) of ( r − 2 ) q = 0 square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) of ...
... conditions . Let 20. Then the boundary conditions are real , and there is exactly one solution ( t , λ ) of ( r − 2 ) q = 0 square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) of ...
Page 1321
... boundary conditions defining T. PROOF . We have seen in the derivation of Theorem 8 that the functions a , ( t ) and ẞ , ( t ) are uniquely determined by the jump equa- tions and by the boundary conditions B * ( K ) = 0 , i = 1 ...
... boundary conditions defining T. PROOF . We have seen in the derivation of Theorem 8 that the functions a , ( t ) and ẞ , ( t ) are uniquely determined by the jump equa- tions and by the boundary conditions B * ( K ) = 0 , i = 1 ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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A₁ 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero