## Linear Operators: General theory |

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Page 630

As may be expected from the attention we have devoted to Theorem 13,

Corollary 14, and Corollary 16, it is important in the present theory to discover

when a closed operator A is the generator of a strongly continuous

Theorem 13 ...

As may be expected from the attention we have devoted to Theorem 13,

Corollary 14, and Corollary 16, it is important in the present theory to discover

when a closed operator A is the generator of a strongly continuous

**semi**-**group**.Theorem 13 ...

Page 655

r^') = ^Li-2g->COsf+g-«' defines a strongly continuous

Equivalently, oo T(t)(x, s) = £ e-|nl< if ... «Mf W. * € 36, where Q3(s, t) = 1+2 2n>-

ic_n,< cos ***» *s a strongly continuous

s) ...

r^') = ^Li-2g->COsf+g-«' defines a strongly continuous

**semi**-**group**on [0, oo).Equivalently, oo T(t)(x, s) = £ e-|nl< if ... «Mf W. * € 36, where Q3(s, t) = 1+2 2n>-

ic_n,< cos ***» *s a strongly continuous

**semigroup**on [0, oo). Equivalently, T(t)(x,s) ...

Page 697

Let (S,Z,/x) be a positive measure space and let {r(<!, . . ., tk), tk > 0} be a strongly

measurable

ck, which is independent of the

...

Let (S,Z,/x) be a positive measure space and let {r(<!, . . ., tk), tk > 0} be a strongly

measurable

**semi**-**group**of operators in ... Jo Then there is an absolute constantck, which is independent of the

**semigroup**and independent of f, such that fi(e*m...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero