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

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1 1 2 a 20 a 16 Let N , N2 , ... be a countable

1 1 2 a 20 a 16 Let N , N2 , ... be a countable

**sequence**of normal operators in ý , all commuting with each other . Show that there exists a single Hermitian operator T such that each N. is a Borel function of T. ( Hint : Use Theorem ...Page 959

Since Ueem = e , the

Since Ueem = e , the

**sequence**{ eembn , m 2 1 } is an increasing { 2 1**sequence**of sets whose union is ebn . Since Mo is countably additive on Bo , uosebn ) = limm Mo ( eembn ) 2 k , and so for some m , Mo ( eem ) 2 uolee mbr ) > k - ε ...Page 1124

If En . E are in F and q ( En ) increases to the limit 9 ( E ) , then it follows from what we have already proved that E , is an increasing

If En . E are in F and q ( En ) increases to the limit 9 ( E ) , then it follows from what we have already proved that E , is an increasing

**sequence**of projections and En S E. If Ec is the strong limit of En , then E SE and q ( E ) = 4 ...### 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 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