Linear Operators: Spectral theory |
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Page 1146
... representations . If such a representation acts in a finite dimensional space E " , then introducing a basis for E " , we may regard the representation as being described by a set of unitary matrices { U ,, ( g ) } . The individual ...
... representations . If such a representation acts in a finite dimensional space E " , then introducing a basis for E " , we may regard the representation as being described by a set of unitary matrices { U ,, ( g ) } . The individual ...
Page 1147
... representation of G is equivalent to one of the representations R ( * ) . COROLLARY : If G is a compact topological group satisfying the second axiom of countability , and G is not a finite set , then any complete set of representations ...
... representation of G is equivalent to one of the representations R ( * ) . COROLLARY : If G is a compact topological group satisfying the second axiom of countability , and G is not a finite set , then any complete set of representations ...
Page 1217
... representation of a Hilbert space relative to a self adjoint operator T in $ is said to be an ordered representation of H relative to T. The measure μ is called the measure of the ordered representation . The sets e , will be called the ...
... representation of a Hilbert space relative to a self adjoint operator T in $ is said to be an ordered representation of H relative to T. The measure μ is called the measure of the ordered representation . The sets e , will be called the ...
Contents
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