## Linear Operators: Spectral operators |

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Page 2068

It is clear that the linear space Lo = Lin Lo

C of all integrable functions which coincide almost everywhere with a continuous

function clearly

It is clear that the linear space Lo = Lin Lo

**satisfies**( 11 ) . Also the space Lo = LinC of all integrable functions which coincide almost everywhere with a continuous

function clearly

**satisfies**( 11 ) . Example 11 . 12 shows that the space Lo = Lin ...Page 2112

The proof of this theorem is based on an analysis of certain B - spaces of vector

valued analytic functions . We say that T

set U and every sequence { fr } of analytic functions from U to X and X e X such ...

The proof of this theorem is based on an analysis of certain B - spaces of vector

valued analytic functions . We say that T

**satisfies**condition ( B ) if for every openset U and every sequence { fr } of analytic functions from U to X and X e X such ...

Page 2385

By Lemma VII . 6 . 1 , ( I – Ku ) - 1 exists , and by Lemmas VII . 6 . 3 and VII . 6 . 4 ,

the vector fu e C [ a , co ) defined by fu = ( I – Kw ) - 11

and is continuous in u for u E Pat and analytic in u for u interior to P # . We have ...

By Lemma VII . 6 . 1 , ( I – Ku ) - 1 exists , and by Lemmas VII . 6 . 3 and VII . 6 . 4 ,

the vector fu e C [ a , co ) defined by fu = ( I – Kw ) - 11

**satisfies**' ful < 2 , u e P + ,and is continuous in u for u E Pat and analytic in u for u interior to P # . We have ...

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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