## Linear Operators: Spectral operators |

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Page 2232

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

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

of the

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

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

of the

**preceding**theorem it follows that E ( e ) D ( Q ) & D ( Q ) and QE ( e ) .Page 2396

Let o4 be as in the

) = A ( 04 ( :, pla ) ) ) ( cf. Lemma 4 for the definition of ula ) ) . Then , by the

) / as ...

Let o4 be as in the

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

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

Page 2455

Therefore , if we let the three operators of the

obtain the present corollary . Q.E.D. 5 COROLLARY . Under the hypotheses of

the

onto ...

Therefore , if we let the three operators of the

**preceding**lemma be H2 , H , H , weobtain the present corollary . Q.E.D. 5 COROLLARY . Under the hypotheses of

the

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

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

32 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero