Linear Operators: Spectral operators |
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Page 2169
This shows that ( vi ) holds for every bounded Borel function f and every
continuous function g . A repetition of this argument shows that it also holds if f
and g are both bounded Borel functions . Thus the operators f ( T ) and g ( T )
commute and ...
This shows that ( vi ) holds for every bounded Borel function f and every
continuous function g . A repetition of this argument shows that it also holds if f
and g are both bounded Borel functions . Thus the operators f ( T ) and g ( T )
commute and ...
Page 2239
Since f xe is a bounded function , the operator Tlf xe ) is a bounded operator . If x
is in E ( @ ) X as well as in E ( e ) X , it follows from the operational calculus for
bounded functions ( cf . XVII . 2 . 10 ) that TƯxe ) 2 = TƯxe ) ( + ) x = TƯxee ) ?
Since f xe is a bounded function , the operator Tlf xe ) is a bounded operator . If x
is in E ( @ ) X as well as in E ( e ) X , it follows from the operational calculus for
bounded functions ( cf . XVII . 2 . 10 ) that TƯxe ) 2 = TƯxe ) ( + ) x = TƯxee ) ?
Page 2252
An attempt to follow the development in the bounded case by writing N = T - S
runs into difficulties . The operator N , although easily seen by Lemma 2 to have a
quasi - nilpotent restriction to each space E ( 0 ) X with o bounded , need not be ...
An attempt to follow the development in the bounded case by writing N = T - S
runs into difficulties . The operator N , although easily seen by Lemma 2 to have a
quasi - nilpotent restriction to each space E ( 0 ) X with o bounded , need not be ...
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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