Linear Operators: Spectral theory |
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Page 888
... set . Here we have used the notations A B and A v B for the intersection and union of two commuting projections A ... spectral sets onto a Boolean algebra of projection operators in X and that furthermore this homomorphism takes the unit σ ( ...
... set . Here we have used the notations A B and A v B for the intersection and union of two commuting projections A ... spectral sets onto a Boolean algebra of projection operators in X and that furthermore this homomorphism takes the unit σ ( ...
Page 933
... spectra . The spectral sets of von Neumann . If T is a bounded linear operator in a Hilbert space , then von Neumann [ 3 ] defines a closed set S of the complex sphere to be a spectral set of T if f ( T ) exists and \ f ( T ) | ≤1 ...
... spectra . The spectral sets of von Neumann . If T is a bounded linear operator in a Hilbert space , then von Neumann [ 3 ] defines a closed set S of the complex sphere to be a spectral set of T if f ( T ) exists and \ f ( T ) | ≤1 ...
Page 993
... spectral set consisting of the single point m then , for some complex number a , q ( x ) = x [ x , m ] for almost all x in R. PROOF . In view of Lemma 11 ( d ) it suffices to prove the theorem in the case m = 0. In this case the ...
... spectral set consisting of the single point m then , for some complex number a , q ( x ) = x [ x , m ] for almost all x in R. PROOF . In view of Lemma 11 ( d ) it suffices to prove the theorem in the case m = 0. In this case the ...
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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