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 uniqueness ...
... 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 uniqueness ...
Page 1152
... 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 ( Ex ) ...
... 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 ( Ex ) ...
Page 1154
... measure in R. Then the product measure λ × λ is a Haar measure in R × R. = PROOF . Since the product group R ( 2 ) RR is locally compact and o - compact , it has a Haar measure ( 2 ) defined on its Borel field ( 2 ) and what we shall ...
... measure in R. Then the product measure λ × λ is a Haar measure in R × R. = PROOF . Since the product group R ( 2 ) RR is locally compact and o - compact , it has a Haar measure ( 2 ) defined on its Borel field ( 2 ) and what we shall ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero