Linear Operators: Spectral theory |
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Page 888
... projections are again projection operators . Also the ranges of the intersection and union of two commuting projection operators are given by the equations ( AB ) X = ( 4X ) ~ ( BX ) , and ( Av B ) X = ( 4X ) + ( BX ) = sp ( AX , BX ) ...
... projections are again projection operators . Also the ranges of the intersection and union of two commuting projection operators are given by the equations ( AB ) X = ( 4X ) ~ ( BX ) , and ( Av B ) X = ( 4X ) + ( BX ) = sp ( AX , BX ) ...
Page 1126
... projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of the projections E ,, it follows from ( 1 ) that the closure in ( a ) of the vectors ( 4 ) is ( m ) . Thus ...
... projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of the projections E ,, it follows from ( 1 ) that the closure in ( a ) of the vectors ( 4 ) is ( m ) . Thus ...
Page 1777
... projection E in § is called an orthogonal projection if the manifolds ES and ( 1 - E ) are orthogonal . = It has been shown in Lemma 4 that = M → ( HM ) where M is an arbitrary closed linear manifold in H. If x = y + z where yЄM and ...
... projection E in § is called an orthogonal projection if the manifolds ES and ( 1 - E ) are orthogonal . = It has been shown in Lemma 4 that = M → ( HM ) where M is an arbitrary closed linear manifold in H. If x = y + z where yЄM and ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero