Linear Operators: Spectral theory |
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Page 950
... measure is unique up to multiplication by positive numbers , and is called Haar measure . In the case R = ( —∞ , ∞ ) , the Haar measure may be taken to be Lebesgue measure : in the case of a compact group , its existence and ...
... measure is unique up to multiplication by positive numbers , and is called Haar measure . In the case R = ( —∞ , ∞ ) , the Haar measure may be taken to be Lebesgue measure : in the case of a compact group , its existence and ...
Page 1152
... Haar measure which , though elementary , are not obvious consequences of the invariance property . 4 LEMMA . Let R be a locally compact , o - compact , Abelian topological group , E its Borel field , and λ its Haar measure . Then λ ( E ...
... Haar measure which , though elementary , are not obvious consequences of the invariance property . 4 LEMMA . Let R be a locally compact , o - compact , Abelian topological group , E its Borel field , and λ its Haar measure . Then λ ( E ...
Page 1154
... Haar measure in R. Then the product measure λ × λ is a Haar measure in R × R. = PROOF . Since the product group R ( 2 ) RXR is locally compact and o - compact , it has a Haar measure ( 2 ) defined on its Borel field ( 2 ) and what we ...
... Haar measure in R. Then the product measure λ × λ is a Haar measure in R × R. = PROOF . Since the product group R ( 2 ) RXR is locally compact and o - compact , it has a Haar measure ( 2 ) defined on its Borel field ( 2 ) and what we ...
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