Linear Operators, Part 2 |
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Page 888
... spectral sets and where & is the void set . Here we have used the notations A ^ B and A v B for the intersection and ... measure in a B - space X. A spectral measure in is a homomorphic map of a Boolean algebra of sets into a Boolean ...
... spectral sets and where & is the void set . Here we have used the notations A ^ B and A v B for the intersection and ... measure in a B - space X. A spectral measure in is a homomorphic map of a Boolean algebra of sets into a Boolean ...
Page 889
... spectral measure satisfying ( iii ) is necessarily an open and closed subset of o ( T ) and thus a spectral set . However , in order to reduce the study of T to its study on invariant subspaces in which it has a smaller spectrum it is ...
... spectral measure satisfying ( iii ) is necessarily an open and closed subset of o ( T ) and thus a spectral set . However , in order to reduce the study of T to its study on invariant subspaces in which it has a smaller spectrum it is ...
Page 897
... spectral measure and , in particular , all of the projections E ( 8 ) commute . It follows then from ( iii ) that the projections E ( 8 ) also commute with T ( f ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The ...
... spectral measure and , in particular , all of the projections E ( 8 ) commute . It follows then from ( iii ) that the projections E ( 8 ) also commute with T ( f ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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Common terms and phrases
adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero