Linear Operators: Spectral operators |
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Page 2299
Hence we find that for n sufficiently large , each u in Cn is in p ( T + P ) and that R
( u ; T + P ) = Blu ) . Since Blu ) is clearly the product of the compact operator R ( u
; T ) and a bounded operator , it follows that T + P is a discrete operator .
Hence we find that for n sufficiently large , each u in Cn is in p ( T + P ) and that R
( u ; T + P ) = Blu ) . Since Blu ) is clearly the product of the compact operator R ( u
; T ) and a bounded operator , it follows that T + P is a discrete operator .
Page 2300
Since E , + Xiai Elan ; T ) = 1 , 1 - Ê has a finite dimensional range for all p . Thus ,
by Lemma VII . 6 . 7 , I - E , has finite dimensional range for all sufficiently large p .
Since E is a countably additive spectral resolution , we have Elu ; T + P ) ( I – E ...
Since E , + Xiai Elan ; T ) = 1 , 1 - Ê has a finite dimensional range for all p . Thus ,
by Lemma VII . 6 . 7 , I - E , has finite dimensional range for all sufficiently large p .
Since E is a countably additive spectral resolution , we have Elu ; T + P ) ( I – E ...
Page 2394
Then there exists a solution oz ( t , u ) of the equation to = u o , defined for 0 St <
oo and for all sufficiently small u € P + , such that og and o ' s are continuous in t
and Me for 0 St < oo and je sufficiently small , and such that ozlt , p ) ~ e - itu ; oślt
...
Then there exists a solution oz ( t , u ) of the equation to = u o , defined for 0 St <
oo and for all sufficiently small u € P + , such that og and o ' s are continuous in t
and Me for 0 St < oo and je sufficiently small , and such that ozlt , p ) ~ e - itu ; oślt
...
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Contents
SPECTRAL OPERATORS | 1924 |
An Operational Calculus for Bounded Spectral | 1941 |
Part | 1950 |
Copyright | |
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adjoint operator analytic apply arbitrary assume B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded operator Chapter clear closed commuting compact complex consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator discrete domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator Math Moreover multiplicity norm 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 uniform uniformly unique valued vector weakly zero