## Linear Operators: Spectral theory |

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Page 1436

Let {g,} be a bounded sequence of elements of £(T) such that {Tg,}

Find a subsequence {g,} = {h} such that r' (h,)

h, = h,-X pro(h)p, is in 3), and Th, – Th, . Thus (h,

Let {g,} be a bounded sequence of elements of £(T) such that {Tg,}

**converges**.Find a subsequence {g,} = {h} such that r' (h,)

**converges**for each j, 1 < j < k. Thenh, = h,-X pro(h)p, is in 3), and Th, – Th, . Thus (h,

**converges**, so that {h} = {h ...Page 1461

Let u > 0; we shall show, using Corollary 2, that —(u-He)go.(t+11). If this is false,

then by Corollary 2 there exists a bounded sequence {f,} of £(To(t+11)) such that

{(t+11+u-He)f,}

Let u > 0; we shall show, using Corollary 2, that —(u-He)go.(t+11). If this is false,

then by Corollary 2 there exists a bounded sequence {f,} of £(To(t+11)) such that

{(t+11+u-He)f,}

**converges**, but such that {fi} has no convergent subsequence.Page 1664

Then the expression F, - F(e-o") is called the Lth Fourier coefficient of F. The

formal series (2+)-" X Freit * L is called the Fourier series of F. 39 LEMMA. The

Fourier series of an element F in D, (C)

follows ...

Then the expression F, - F(e-o") is called the Lth Fourier coefficient of F. The

formal series (2+)-" X Freit * L is called the Fourier series of F. 39 LEMMA. The

Fourier series of an element F in D, (C)

**converges**unconditionally to F. PRoof. Itfollows ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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