## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 909

Spectral Representation Let и be a finite positive measure defined on the

Spectral Representation Let и be a finite positive measure defined on the

**Borel**sets B of the complex plane and vanishing on the complement of a bounded set S. One of the simplest examples of a bounded normal operator is the operator T ...Page 913

PROOF . We can clearly suppose that \ yol 1. Let Yo , 41 , 42 , • • be an orthonormal basis for H , whose initial element is yo . Let E be the spectral resolution for T and let vn ( e ) = ( E ( e ) yn , yn ) for each

PROOF . We can clearly suppose that \ yol 1. Let Yo , 41 , 42 , • • be an orthonormal basis for H , whose initial element is yo . Let E be the spectral resolution for T and let vn ( e ) = ( E ( e ) yn , yn ) for each

**Borel**set e .Page 1900

( See also Boolean ring ) definition , ( 43 ) properties , ( 44 ) representation of , ( 44 ) Boolean ring , definition , ( 40 ) representation of , 1.12.1 ( 41 )

( See also Boolean ring ) definition , ( 43 ) properties , ( 44 ) representation of , ( 44 ) Boolean ring , definition , ( 40 ) representation of , 1.12.1 ( 41 )

**Borel**field of sets , definition , III.5.10 ( 137 )**Borel**function , X.1 ...### What people are saying - Write a review

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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