Linear Operators: Spectral theory |
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Page 1347
... evident advantage that it makes o , ( , λ ) an entire function of the complex variable λ ; but it has drawbacks which , though less evident , are nevertheless decisive . Suppose , for example , that we study the self adjoint operator T ...
... evident advantage that it makes o , ( , λ ) an entire function of the complex variable λ ; but it has drawbacks which , though less evident , are nevertheless decisive . Suppose , for example , that we study the self adjoint operator T ...
Page 1695
... evident that the series ( x ) = j = 1 P ( x ) converges to a function y in C ( I ) which is everywhere positive . Thus , if we put n , ( x ) = y ( x ) -1y , ( x ) , we have n , in Co ( I ) , and ∞ Ση ; ( x ) = 1 . Since only a finite ...
... evident that the series ( x ) = j = 1 P ( x ) converges to a function y in C ( I ) which is everywhere positive . Thus , if we put n , ( x ) = y ( x ) -1y , ( x ) , we have n , in Co ( I ) , and ∞ Ση ; ( x ) = 1 . Since only a finite ...
Page 1759
... evident on repeated partial differentiation of this equation that any derivative Jig of g of order at most k can be expressed as a linear combination of the " pure x - derivatives " g of g of order at most k ; that is , it is evident ...
... evident on repeated partial differentiation of this equation that any derivative Jig of g of order at most k can be expressed as a linear combination of the " pure x - derivatives " g of g of order at most k ; that is , it is evident ...
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