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 1210
... measure space . Let E be the resolution of the identity for T. We assume that there exists an increasing sequence ... Lebesgue measure . We will see that every function in D ( T ) is 1210 XII.3.7 XII . UNBOUNDED OPERATORS IN HILBERT SPACE.
... measure space . Let E be the resolution of the identity for T. We assume that there exists an increasing sequence ... Lebesgue measure . We will see that every function in D ( T ) is 1210 XII.3.7 XII . UNBOUNDED OPERATORS IN HILBERT SPACE.
Page 1213
... Lebesgue measure , and W will , for almost all 2 , be square in- tegrable over every compact subset of S. In this case the limit F of the preceding definition is independent of the sequence { S } provided that the sets S are compact . n ...
... Lebesgue measure , and W will , for almost all 2 , be square in- tegrable over every compact subset of S. In this case the limit F of the preceding definition is independent of the sequence { S } provided that the sets S are compact . n ...
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