## Linear Operators: Spectral theory |

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Page 1094

Statement (a) is evident from Definition 1. To prove (b), we put T = Ti-HT, and

note that, by Corollary 3, Manil (T1-HT2) is unti(T1)+/1.11(T2) Alan 2(Ti-HT2) is u,

11(T1)+/1,12(T2). First let p > 1. Then by Minkowski's

3.

Statement (a) is evident from Definition 1. To prove (b), we put T = Ti-HT, and

note that, by Corollary 3, Manil (T1-HT2) is unti(T1)+/1.11(T2) Alan 2(Ti-HT2) is u,

11(T1)+/1,12(T2). First let p > 1. Then by Minkowski's

**inequality**, oo 1/p co 1/p co (3.

Page 1105

We now pause to sharpen another of the

the paragraph following Lemma 9, the continuity of the norm function which

follows from the triangle

that ...

We now pause to sharpen another of the

**inequalities**of Lemma 9. ... was noted inthe paragraph following Lemma 9, the continuity of the norm function which

follows from the triangle

**inequality**of Lemma 14(d), and by Lemma 11, it followsthat ...

Page 1774

A Hilbert space $) is a complea B-space and |(a, y) = |a||y|, a, ye Ş. PRoof. The

above

from the postulates for $5 that the Schwarz

zero.

A Hilbert space $) is a complea B-space and |(a, y) = |a||y|, a, ye Ş. PRoof. The

above

**inequality**, known as the Schwarz**inequality**, will be proved first. It followsfrom the postulates for $5 that the Schwarz

**inequality**is valid if either a or y iszero.

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

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additive 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 satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero