Linear Operators: Spectral theory |
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Page 892
... map f → Ss f ( s ) E ( ds ) is a continuous linear map of B ( S , 2 ) into the algebra of bounded operators in X. If the set function E is a spectral measure , the map f → Ss f ( s ) E ( ds ) is also a homomorphism . To see this let ...
... map f → Ss f ( s ) E ( ds ) is a continuous linear map of B ( S , 2 ) into the algebra of bounded operators in X. If the set function E is a spectral measure , the map f → Ss f ( s ) E ( ds ) is also a homomorphism . To see this let ...
Page 1267
... map u → u ( T ) is an order - preserving linear map of the real algebra o into the real algebra ( H ) , and that ( Hint : Use Exercise 32. ) 0 u ( T ) max u ( 2 ) . a≤1 ( b ) Extend the homomorphism of part ( a ) to an order - preserv ...
... map u → u ( T ) is an order - preserving linear map of the real algebra o into the real algebra ( H ) , and that ( Hint : Use Exercise 32. ) 0 u ( T ) max u ( 2 ) . a≤1 ( b ) Extend the homomorphism of part ( a ) to an order - preserv ...
Page 1299
... linear mapping of the space of boundary values for t ' at a onto the space ... linear operator defined by the equation In the formula = ( S1f ) ( t ) n n ' k ... map from M ' to M. It will be shown that Ø1 is one - to - one and that Ø1 ...
... linear mapping of the space of boundary values for t ' at a onto the space ... linear operator defined by the equation In the formula = ( S1f ) ( t ) n n ' k ... map from M ' to M. It will be shown that Ø1 is one - to - one and that Ø1 ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero