## Linear Operators: Spectral operators |

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Page 2232

Since Q is closed , it follows that x is in D ( Q ) and that Qx = Qx . Thus Q SQ . It

follows from symmetry that Q = Q . Q . E . D . 7 COROLLARY . Under the

hypotheses of the

QE ( e ) x ...

Since Q is closed , it follows that x is in D ( Q ) and that Qx = Qx . Thus Q SQ . It

follows from symmetry that Q = Q . Q . E . D . 7 COROLLARY . Under the

hypotheses of the

**preceding**theorem it follows that E ( e ) D ( Q ) & D ( Q ) andQE ( e ) x ...

Page 2396

Let 04 be as in the

) = A ( 04 ( : , M ( A ) ) ) ( cf . Lemma 4 for the definition of u ( a ) ) . Then , by the

...

Let 04 be as in the

**preceding**lemma , put A ( a ) = A ( 0 ( : , f ( a ) ) ) , and let B ( A) = A ( 04 ( : , M ( A ) ) ) ( cf . Lemma 4 for the definition of u ( a ) ) . Then , by the

**preceding**lemma , by Lemma 1 , and by formulas ( 2a ) and ( 2b ) , ( B ( 2 ) ~ | A )...

Page 2455

Therefore , if we let the three operators of the

we obtain the present corollary . Q . E . D . 5 COROLLARY . Under the

hypotheses of the

( H1 , H2 ) ...

Therefore , if we let the three operators of the

**preceding**lemma be H2 , H1 , H1 ,we obtain the present corollary . Q . E . D . 5 COROLLARY . Under the

hypotheses of the

**preceding**corollary , U ( H1 , H2 ) is an isometric mapping of E( H1 , H2 ) ...

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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