## Linear Operators: Spectral operators |

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Page 2264

section are due to Bade and are intimately dependent upon the ideas introduced

in Section XVII . 3 . A

...

**Multiplicity**Theory and Spectral Representation The methods and results of thissection are due to Bade and are intimately dependent upon the ideas introduced

in Section XVII . 3 . A

**multiplicity**theory for Boolean algebras of projections in a B...

Page 2283

Then a projection E in B has finite uniform

in B * has finite uniform

* satisfy the countable chain condition . Also since each projection is the union ...

Then a projection E in B has finite uniform

**multiplicity**n if and only if its adjoint E *in B * has finite uniform

**multiplicity**n . PROOF . It is sufficient to suppose E and E* satisfy the countable chain condition . Also since each projection is the union ...

Page 2289

projections of finite

undecided even whether E * has infinite uniform

uniform

projections of finite

**multiplicity**. We have no information on this question . It isundecided even whether E * has infinite uniform

**multiplicity**when E has infiniteuniform

**multiplicity**. A related question concerns invariant subspaces . If M is an ...### What people are saying - Write a review

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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