Linear Operators: Spectral theory |
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Page 1310
... solution y ( t , λ ) of ( x - 2 ) y = 0 square - integrable at b and satisfying the boundary conditions at b . PROOF ... solution y of ( 7-2 ) = 0 square - integrable at b and satisfying all boundary conditions at b , and at least one ...
... solution y ( t , λ ) of ( x - 2 ) y = 0 square - integrable at b and satisfying the boundary conditions at b . PROOF ... solution y of ( 7-2 ) = 0 square - integrable at b and satisfying all boundary conditions at b , and at least one ...
Page 1472
... solution of tσ = λσ square- integrable at a and satisfying the boundary condition B , so is f . Since , by the preceding lemma , only one such solution ( up to a constant multiple ) exists , we must have ƒ af , where , since || = | ƒ ...
... solution of tσ = λσ square- integrable at a and satisfying the boundary condition B , so is f . Since , by the preceding lemma , only one such solution ( up to a constant multiple ) exists , we must have ƒ af , where , since || = | ƒ ...
Page 1632
... solution for each set of prescribed , smooth initial data . Property B : The solutions of Lf = 0 are so smooth as to be subject to the function - theoretic principle of unique continuation . Formal partial differential operators with ...
... solution for each set of prescribed , smooth initial data . Property B : The solutions of Lf = 0 are so smooth as to be subject to the function - theoretic principle of unique continuation . Formal partial differential operators with ...
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