Linear Operators: Spectral theory |
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Page 878
... corresponding to the function f in C ( σ ( x ) ) under the * -isomorphism between B * ( x ) and C ( o ( x ) ) which is uniquely defined by the require- ment that x and g ( u ) = μ be corresponding elements . μ The notation introduced in ...
... corresponding to the function f in C ( σ ( x ) ) under the * -isomorphism between B * ( x ) and C ( o ( x ) ) which is uniquely defined by the require- ment that x and g ( u ) = μ be corresponding elements . μ The notation introduced in ...
Page 942
... corresponding to λ is also an eigenfunction corresponding to 2. Thus every eigenfunction of T , which corresponds to a non - zero eigenvalue is a finite dimensional continuous function . Hence N is orthogonal to every eigenfunction of T ...
... corresponding to λ is also an eigenfunction corresponding to 2. Thus every eigenfunction of T , which corresponds to a non - zero eigenvalue is a finite dimensional continuous function . Hence N is orthogonal to every eigenfunction of T ...
Page 1729
... corresponding case of the space Ca ( C ) , we may regard any point x = [ x1 , y ] for which 0 < x < 27 as belonging , in a suitable sense , to the interior of C ; that is , to argue at such a point as we would at an interior point , we ...
... corresponding case of the space Ca ( C ) , we may regard any point x = [ x1 , y ] for which 0 < x < 27 as belonging , in a suitable sense , to the interior of C ; that is , to argue at such a point as we would at an interior point , we ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero