## Linear Operators: Spectral operators |

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Page 2144

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

show that A ( 0 ) A ( 8 ) = A ( od ) . It is seen , by using the above remarks , that for

...

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

**topology**of X * , and is bounded . It remains only toshow that A ( 0 ) A ( 8 ) = A ( od ) . It is seen , by using the above remarks , that for

...

Page 2203

The mapping E HO ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

The mapping E HO ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

**topology**in 1 . To see this , note that sets of the form { a | | ( T ...Page 2319

Then , if f is in the domain D ( T ) , the series expansion ; T ) f converges to f

unconditionally in the

certainly converges unconditionally in the

, so ...

Then , if f is in the domain D ( T ) , the series expansion ; T ) f converges to f

unconditionally in the

**topology**of A ( n ) ( J ) . PROOF . The series 1 E ( Ni ; T ) fcertainly converges unconditionally in the

**topology**of L2 ( J ) . On the other hand, so ...

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algebra of projections analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear closure commuting compact complete consider constant contained 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 positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero