Linear Operators, Part 2 |
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Page 1327
... obtained from 7 by imposing a real symmetric set of boundary con- ditions , a further simplification can be obtained by noticing that the solutions of the equation ( τ * —Ã ) σ = 0 are complex conjugates of the solutions of the ...
... obtained from 7 by imposing a real symmetric set of boundary con- ditions , a further simplification can be obtained by noticing that the solutions of the equation ( τ * —Ã ) σ = 0 are complex conjugates of the solutions of the ...
Page 1617
... obtained by the imposition of separated boundary conditions , and let { g } be the orthonormal eigenfunctions obtained from the operator - ( d / dt ) 2 by the imposition of the same boundary conditions on an interval [ a , b ] . Let f ...
... obtained by the imposition of separated boundary conditions , and let { g } be the orthonormal eigenfunctions obtained from the operator - ( d / dt ) 2 by the imposition of the same boundary conditions on an interval [ a , b ] . Let f ...
Page 1619
... obtained by the imposition of a boundary condition at zero will consist of a continuous spectrum covering the positive real semi - axis , and a sequence of eigenvalues with simple normalized eigenfunctions { f } . Let f ( t , 2 ) be a ...
... obtained by the imposition of a boundary condition at zero will consist of a continuous spectrum covering the positive real semi - axis , and a sequence of eigenvalues with simple normalized eigenfunctions { f } . Let f ( t , 2 ) be a ...
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 domain 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 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 tr(T transform unique unitary vanishes vector zero