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

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Page 889

A spectral measure E defined on the

A spectral measure E defined on the

**Borel sets**in the plane and satisfying ( iv ) for every**Borel set**8 and ( v ) ... of disjoint**Borel sets**is called a resolution of the identity for T. With this terminology the spectral theorem for ...Page 894

6.2 ) that x * E ( ) . x = 0 for every

6.2 ) that x * E ( ) . x = 0 for every

**Borel set**d in S and 8 ) every pair x , æ * with x e X , ** EX * . It follows ( II.3.15 ) that E ( S ) 0 . Thus if E and A are bounded additive regular operator valued set functions defined on the ...Page 913

Let E be the spectral resolution for T and let vn ( e ) = ( E ( e ) yn , yn ) for each

Let E be the spectral resolution for T and let vn ( e ) = ( E ( e ) yn , yn ) for each

**Borel set**e . ... let { en } be a sequence } of**Borel sets**such that X = 7 v ; ( en ) = 0 , and such that if e is a Borel subset of the complement en ...### 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