Linear Operators: Spectral theory |
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Page 1433
... equation which have real parts greater than - ( 1/2 ) . In case the indicial equation has two roots which differ by an integer or has multiple roots , a corresponding result may be stated ; but here the basis for solutions of [ * ] may ...
... equation which have real parts greater than - ( 1/2 ) . In case the indicial equation has two roots which differ by an integer or has multiple roots , a corresponding result may be stated ; but here the basis for solutions of [ * ] may ...
Page 1527
... equation we must consequently use some of the theory of equations with irregular singular points . It is most convenient to take the equation Lf ( d / dz ) 2f + p ( z ) ( d / dz ) f + q ( z ) f = 0 to have an irregular singularity of ...
... equation we must consequently use some of the theory of equations with irregular singular points . It is most convenient to take the equation Lf ( d / dz ) 2f + p ( z ) ( d / dz ) f + q ( z ) f = 0 to have an irregular singularity of ...
Page 1528
... equations for the coefficients . The first of these algebraic equations , which is simply the characteristic equation of the differential equation , is quadratic ; all the succeeding equations are linear . If we find the differential ...
... equations for the coefficients . The first of these algebraic equations , which is simply the characteristic equation of the differential equation , is quadratic ; all the succeeding equations are linear . If we find the differential ...
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