## Linear Operators: Spectral theory |

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Page 1036

i-1 A

T) is a continuous complew valued function on the B-space of all Hilbert-Schmidt

operators. PRoof. First note that if t is a complex number with |t| < 1 then (+) log ...

i-1 A

**converges**and defines a function analytic for A # 0. For each fired A # 0, pa(T) is a continuous complew valued function on the B-space of all Hilbert-Schmidt

operators. PRoof. First note that if t is a complex number with |t| < 1 then (+) log ...

Page 1420

Suppose that {f}

assumption (b), {f,}

{f,}

{f} is ...

Suppose that {f}

**converges**to zero in the topology of 3)(Ti(r)). Then, byassumption (b), {f,}

**converges**to zero in the topology of?)(Ti(t-i-t')). Conversely, let{f,}

**converge**to zero in the topology of $(Ti(t-i-t')), that is, let [+] fal+|Ti(t-Ht')f, -> 0. If{f} is ...

Page 1436

Let {gn} be a bounded sequence of elements of 3)(T) such that (Tga}

Find a subsequence {gr} = {h} such that af(h,)

h, → h, →X; it?(h,)p, is in 3), and Th, – Th,. Thus (h,}

Let {gn} be a bounded sequence of elements of 3)(T) such that (Tga}

**converges**.Find a subsequence {gr} = {h} such that af(h,)

**converges**for each j, 1 < j < k. Thenh, → h, →X; it?(h,)p, is in 3), and Th, – Th,. Thus (h,}

**converges**, so that {h} = {h ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 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