Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |
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Page 1152
The existence of an invariant measure on a group satisfying the second axiom of countability was first shown by Haar [ 1 ] ... Other results concerning measures invariant under transformations are found in Oxtoby and Ulam [ 1 ] .
The existence of an invariant measure on a group satisfying the second axiom of countability was first shown by Haar [ 1 ] ... Other results concerning measures invariant under transformations are found in Oxtoby and Ulam [ 1 ] .
Page 1153
2 Since the measure space ( R , E , a ) is a o - finite measure space the theory of integration as developed in Chapter III may be used as a basis for the theory developed in Sections 3-4 . In particular we should notice that the ...
2 Since the measure space ( R , E , a ) is a o - finite measure space the theory of integration as developed in Chapter III may be used as a basis for the theory developed in Sections 3-4 . In particular we should notice that the ...
Page 1154
o - compact group R and let à be a Haar measure in R. Then the product measure 2 xa is a Haar measure in RX R. PROOF . Since the product group R ( 2 ) = R XR is locally compact Rx and o - compact , it has a Haar measure 2 ( 2 ) defined ...
o - compact group R and let à be a Haar measure in R. Then the product measure 2 xa is a Haar measure in RX R. PROOF . Since the product group R ( 2 ) = R XR is locally compact Rx and o - compact , it has a Haar measure 2 ( 2 ) defined ...
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additive adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined eigenvalues element equal equation Exercise exists extension fact finite dimensional follows formal formal differential operator formula function function f given Hence Hilbert space Hilbert-Schmidt ideal identity independent inequality integral interval isometric isomorphism Lemma limit linear matrix measure multiplicity neighborhood norm normal operator obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solution spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero