Linear Operators: Spectral theory |
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Page 1310
... square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) of ( t −2 ) y square - integrable at b and satisfying the boundary conditions at b . - 0 PROOF . We shall show the theorem is ...
... square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) of ( t −2 ) y square - integrable at b and satisfying the boundary conditions at b . - 0 PROOF . We shall show the theorem is ...
Page 1321
... square - integrable in a neighborhood of b . In the same way it may be shown that the functions ẞ , are square- integrable in a neighborhood of a . Q.E.D. The following corollary enables us to use Theorem 8 as a com- putational ...
... square - integrable in a neighborhood of b . In the same way it may be shown that the functions ẞ , are square- integrable in a neighborhood of a . Q.E.D. The following corollary enables us to use Theorem 8 as a com- putational ...
Page 1329
... square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) of ( T - 2 ) σ = 0 square- integrable at b satisfying the boundary conditions at b . The resolvent R ( λ ; T ) is an integral ...
... square - integrable at a and satisfying the boundary conditions at a , and exactly one solution y ( t , λ ) of ( T - 2 ) σ = 0 square- integrable at b satisfying the boundary conditions at b . The resolvent R ( λ ; T ) is an integral ...
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