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

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Page 1931

If x is a vector in X , then by an analytic

X - valued function f defined and analytic on an open set D ( F ) 2 P ( T ) and such

that ( I – T ) $ ( $ ) = x , Ś € D ( f ) . It is clear that , for such an

If x is a vector in X , then by an analytic

**extension**of R ( E ; T ' ) x will be meant anX - valued function f defined and analytic on an open set D ( F ) 2 P ( T ) and such

that ( I – T ) $ ( $ ) = x , Ś € D ( f ) . It is clear that , for such an

**extension**, f ...Page 2092

The single valued

not have the single valued

S . Kakutani ( see Dunford [ 18 ] ) . Kesel ' man [ 1 ] gave necessary conditions for

...

The single valued

**extension**property . The example of an operator which doesnot have the single valued

**extension**property that is given in Section 2 is due toS . Kakutani ( see Dunford [ 18 ] ) . Kesel ' man [ 1 ] gave necessary conditions for

...

Page 2095

Consequently , the analogs of the results stated in the preceding paragraph for

restrictions of spectral and scalar type operators also hold for their quotients .

Although the restrictions of operators with the single valued

have ...

Consequently , the analogs of the results stated in the preceding paragraph for

restrictions of spectral and scalar type operators also hold for their quotients .

Although the restrictions of operators with the single valued

**extension**propertyhave ...

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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