## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 1947

Let B1 , . . . , Bk be a finite collection of commuting bounded Boolean algebras of

projections in a Hilbert space H . Then there exists a bounded self

operator B in H with a bounded everywhere defined inverse such that BEB - 1 is

a self ...

Let B1 , . . . , Bk be a finite collection of commuting bounded Boolean algebras of

projections in a Hilbert space H . Then there exists a bounded self

**adjoint**operator B in H with a bounded everywhere defined inverse such that BEB - 1 is

a self ...

Page 2106

such a mapping y , we say that A e B ( X ) is

equivalently , if ( Ax ) * = A * * * for all x e X ) . If A is

real and A2n is Hermitian for n = 1 , 2 , . . . ; however ; A need not be Hermitian ,

and cI ...

such a mapping y , we say that A e B ( X ) is

**adjoint**Abelian if A * g = qA (equivalently , if ( Ax ) * = A * * * for all x e X ) . If A is

**adjoint**Abelian , then O ( A ) isreal and A2n is Hermitian for n = 1 , 2 , . . . ; however ; A need not be Hermitian ,

and cI ...

Page 2169

Self

how the theory of spectral operators may be applied to yield the classical spectral

theorem in Hilbert space , that is , the theorem asserting that a bounded self ...

Self

**Adjoint**Operators in Hilbert Space It is the purpose of this section to showhow the theory of spectral operators may be applied to yield the classical spectral

theorem in Hilbert space , that is , the theorem asserting that a bounded self ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 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