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 ( A ^ B ) X = ( AX ) ~ ( BX ) , and ( A v B ) X = ( 4X ) + ( BX ) = sp ( 4X , BX ) ...
... projections are again projection operators . Also the ranges of the intersection and union of two commuting projection operators are given by the equations ( A ^ B ) X = ( AX ) ~ ( BX ) , and ( A v B ) X = ( 4X ) + ( BX ) = sp ( 4X , BX ) ...
Page 1123
... projection . We say that E is a subdiagonalizing projection for T if T leaves the range of E invariant , i.e. , if ETE TE . - 3 LEMMA . Any operator T in Hilbert space admits a maximal totally ordered set F of orthogonal ...
... projection . We say that E is a subdiagonalizing projection for T if T leaves the range of E invariant , i.e. , if ETE TE . - 3 LEMMA . Any operator T in Hilbert space admits a maximal totally ordered set F of orthogonal ...
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 ( m ) 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 ( m ) of the vectors ( 4 ) is ( m ) . Thus ...
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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm 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 transform unique unitary vanishes vector zero