Linear Operators: Spectral theory |
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Page 888
... projection operators are given by the equations ( 4 ^ B ) X = ( AX ) ~ ( BX ) , and ( A v B ) X ( 4X ) + ( BX ) = sp ( AX , BX ) , respectively . Thus , in a Boolean algebra of projections , the order relation A≤ B , which is defined ...
... projection operators are given by the equations ( 4 ^ B ) X = ( AX ) ~ ( BX ) , and ( A v B ) X ( 4X ) + ( BX ) = sp ( AX , BX ) , respectively . Thus , in a Boolean algebra of projections , the order relation A≤ B , which is defined ...
Page 924
... algebra A of operators on admits no non - trivial invariant subspaces then A B ( S ) . ( Hint : Use the second ... projections , then E1 ≥ E , in the sense of Section 4 if and only if E , E2 in the sense of Definition VI.3.4 . 1 13 If N is a ...
... algebra A of operators on admits no non - trivial invariant subspaces then A B ( S ) . ( Hint : Use the second ... projections , then E1 ≥ E , in the sense of Section 4 if and only if E , E2 in the sense of Definition VI.3.4 . 1 13 If N is a ...
Page 1128
... algebra of operators generated by the projections E , and let Ʌ be its spectrum . If is any element of Д , i.e. ... algebra A , so that E ( e ) is a countably additive regular projection - valued Borel measure defined on 4 ( cf. Theorem ...
... algebra of operators generated by the projections E , and let Ʌ be its spectrum . If is any element of Д , i.e. ... algebra A , so that E ( e ) is a countably additive regular projection - valued Borel measure defined on 4 ( cf. Theorem ...
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