## Linear Operators: Spectral theory |

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Page 1207

It is clear from Zorn ' s lemma that there is a maximal set A in H for which the

spaces Ha , a € A , are

that no x + 0 is

It is clear from Zorn ' s lemma that there is a maximal set A in H for which the

spaces Ha , a € A , are

**orthogonal**. Thus to prove the lemma it suffices to observethat no x + 0 is

**orthogonal**to each of the spaces Hq . Indeed , if x # 0 is**orthogonal**...Page 1227

Nelson Dunford, Jacob T. Schwartz. numbers ) denoted by nt and n _ , are called

the positive and negative deficiency indices of T , respectively . 10 LEMMA . ( a )

D ( T ) , Dr , and D _ are closed

Nelson Dunford, Jacob T. Schwartz. numbers ) denoted by nt and n _ , are called

the positive and negative deficiency indices of T , respectively . 10 LEMMA . ( a )

D ( T ) , Dr , and D _ are closed

**orthogonal**subspaces of the Hilbert space D ( T ...Page 1262

Then there exists a Hilbert space H , 2H , and an

such that Ax = PQx , XEH , P denoting the

Let { Tn } be a sequence of bounded operators in Hilbert space H . Then there

exists ...

Then there exists a Hilbert space H , 2H , and an

**orthogonal**projection Q in H ,such that Ax = PQx , XEH , P denoting the

**orthogonal**projection of Hi on H . 29Let { Tn } be a sequence of bounded operators in Hilbert space H . Then there

exists ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint operator Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero