## Linear Operators: Spectral operators |

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Page 1979

It follows from equations ( iv ) and ( v ) of Lemma 3 that E ( 1 , ( s ) ; Â ( s ) ) is e -

essentially bounded on S . Lemma 4 then

is satisfied . Q . E . D . 8 COROLLARY . Every operator A in AP is the strong limit ...

It follows from equations ( iv ) and ( v ) of Lemma 3 that E ( 1 , ( s ) ; Â ( s ) ) is e -

essentially bounded on S . Lemma 4 then

**shows**that condition ( i ) of the theoremis satisfied . Q . E . D . 8 COROLLARY . Every operator A in AP is the strong limit ...

Page 2169

This

continuous function g . A repetition of this argument

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 everycontinuous function g . A repetition of this argument

**shows**that it also holds if fand g are both bounded Borel functions . Thus the operators f ( T ) and g ( T )

commute and ...

Page 2170

These lemmas will

expansion of the scalar product ( ( QI – T ) x , ( al – T ' ) x )

2 = \ I ( « ) x12 + | ( R ( Q ) I – T ) x12 2 I ( a ) | 2 | 2 / , so that Ixl slal – T ' ) x1 | I ...

These lemmas will

**show**that the hypotheses of Theorem 5 . ... If a is not real , anexpansion of the scalar product ( ( QI – T ) x , ( al – T ' ) x )

**shows**that llal – T ' ) x |2 = \ I ( « ) x12 + | ( R ( Q ) I – T ) x12 2 I ( a ) | 2 | 2 / , so that Ixl slal – T ' ) x1 | I ...

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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