## Linear Operators: General theory |

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Results 1-3 of 32

Page 613

ta) is the most general continuous real (or complex) function / of the non-

negative real variable t which satisfies the functional equations /(0) = 1, /(/+«) = f{t

)f{u).

**Semi**-**groups**of Operators It is well known that the exponential function /(<) = exp(ta) is the most general continuous real (or complex) function / of the non-

negative real variable t which satisfies the functional equations /(0) = 1, /(/+«) = f{t

)f{u).

Page 655

1 f- (l-er*)x(,+£)de defines a strongly continuous

Equivalently, 00 T(t)(x, a) = Jte-Mtx„ein' if 00 1 C" xls) ~ 2jxneXM, x. = — a;(s)c~""<

Zs. 18 Let X = L„(—7i, n), 1 ^ p < oo. Show that the family of operators defined by

the ...

1 f- (l-er*)x(,+£)de defines a strongly continuous

**semi**-**group**on [0, oo).Equivalently, 00 T(t)(x, a) = Jte-Mtx„ein' if 00 1 C" xls) ~ 2jxneXM, x. = — a;(s)c~""<

Zs. 18 Let X = L„(—7i, n), 1 ^ p < oo. Show that the family of operators defined by

the ...

Page 685

T(t)dt, 0 ^ a, a Jo which are to be our basic concern in this section, it will be

desirable to supplement the algebraic condition (i) by a condition concerning the

analytical dependence of the

...

T(t)dt, 0 ^ a, a Jo which are to be our basic concern in this section, it will be

desirable to supplement the algebraic condition (i) by a condition concerning the

analytical dependence of the

**semi**-**group**upon the parameter t. We recall that the...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero