## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2154

Thus , since the

T – XI ) " X $ ( T – XI ) n + 1 % . ... it follows that ( T – XI ) n + 1 % + { x | ( T – 27 ) n

+ 1x = 0 } is dense in X . By induction , it is seen that the

Thus , since the

**manifolds**( T - XI ) MX decrease as m increases , it follows that (T – XI ) " X $ ( T – XI ) n + 1 % . ... it follows that ( T – XI ) n + 1 % + { x | ( T – 27 ) n

+ 1x = 0 } is dense in X . By induction , it is seen that the

**manifold**( T — XI ) n + ...Page 2214

Then an operator is in the weakly closed algebra generated by B if and only if it

leaves invariant every closed linear

member of B . PROOF . It was observed in the preceding proof that an operator in

the ...

Then an operator is in the weakly closed algebra generated by B if and only if it

leaves invariant every closed linear

**manifold**which is invariant under everymember of B . PROOF . It was observed in the preceding proof that an operator in

the ...

Page 2217

Let X ( x ) , Xy ( 2 ) be the closed linear

} , { Ex Ee Bl } , respectively . ... space if and only if it leaves invariant every closed

linear

Let X ( x ) , Xy ( 2 ) be the closed linear

**manifolds**generated by the sets { EX Ee B} , { Ex Ee Bl } , respectively . ... space if and only if it leaves invariant every closed

linear

**manifold**which remains invariant under every element of B . PROOF .### What people are saying - Write a review

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adjoint operator analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal perturbation plane positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero