Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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 union of two commuting projections A and B. We recall that these operators are defined by the equations A ^ B = AB , A v B ...
... spectral sets and where is the void set . Here we have used the notations A B and A v B for the intersection and union of two commuting projections A and B. We recall that these operators are defined by the equations A ^ B = AB , A v B ...
Page 889
... spectrum of an operator is always closed ( IX.1.5 ) , every set in the domain of a 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 ...
... spectrum of an operator is always closed ( IX.1.5 ) , every set in the domain of a 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 ...
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 ƒ ( 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 ƒ ( T ) | ≤1 ...
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