Linear Operators, Part 2 |
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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 , Σ 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 , Σ its Borel field , and λ its Haar measure . Then λ ( Ex ) ...
Page 1153
... measure ( 2 ) defined on its Borel field ( 2 ) . It is natural to ex- pect that the product measure 2 × λ coincides , up to a constant multi- ple , with 2 ( 2 ) . This fact will be established in Theorem 7 . 6 LEMMA . Let the locally ...
... measure ( 2 ) defined on its Borel field ( 2 ) . It is natural to ex- pect that the product measure 2 × λ coincides , up to a constant multi- ple , with 2 ( 2 ) . This fact will be established in Theorem 7 . 6 LEMMA . Let the locally ...
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 ) = R R is locally compact and σ - compact , it has a Haar measure ( 2 ) defined on its Borel field Σ ( 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 ) = R R is locally compact and σ - compact , it has a Haar measure ( 2 ) defined on its Borel field Σ ( 2 ) and what we shall ...
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 domain 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 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 open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma 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