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

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Page 898

If E is the resolution of the

If E is the resolution of the

**identity**for the normal operator T and if d is a Borel set of complex numbers , then E ( ) T = TE ( S ) , 0 ( T8 ) CJ , where To is the restriction of T to Ed ) .V . Proof . The first statement follows from ...Page 920

Let E and be the resolutions of the

Let E and be the resolutions of the

**identity**for T and I respectively . From Corollary 2.7 it is seen that Ể = VEV - 1 and hence that F ( T ) = VF ( T ) V - 1 for every bounded Borel function F. The mapping W = ŪV of onto Im = Lelēm ...Page 1717

By induction on Jil , we can readily show that a formal

By induction on Jil , we can readily show that a formal

**identity**( 1 ) 201C ( x ) 212 = C ( x ) 201213 + Σ C1,1 ... Making use of**identities**of the type ( 1 ) , we may evidently proceed to prove by induction on the order of t that t may ...### What people are saying - Write a review

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### Other editions - View all

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

### Common terms and phrases

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 Ly(R 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 unique unit unitary vanishes vector zero