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 ( 4 ^ B ) X = ( AX ) ~ ( BX ) , and ( A v 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 ( 4 ^ B ) X = ( AX ) ~ ( BX ) , and ( A v B ) X ( 4X ) + ( BX ) = sp ( AX , BX ) ...
Page 1123
... projection for T if T leaves the range of E invariant , i.e. , if ETE TE . = 3 LEMMA . Any operator T in Hilbert ... projection E. If E , E , are in F , ∞ 12 1 and q ( E ) = q XI.10.2 1123 SUBDIAGONALIZATION OF COMPACT OPERATORS.
... projection for T if T leaves the range of E invariant , i.e. , if ETE TE . = 3 LEMMA . Any operator T in Hilbert ... projection E. If E , E , are in F , ∞ 12 1 and q ( E ) = q XI.10.2 1123 SUBDIAGONALIZATION OF COMPACT OPERATORS.
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 ( am ) . 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 ( am ) . 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 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