## Linear Operators, Part 2 |

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Page 1175

<e> = Tm (s)Then fl is a bounded

. For each real $0, let .9?“ be the

) <12:/><:> vs). e > so. otherwise. ll ll Q *1 By Corollary 22, it follows that there is ...

<e> = Tm (s)Then fl is a bounded

**mapping**of the space L¢(L,(S)) into itself. Pnoor. For each real $0, let .9?“ be the

**mapping**in Lq(L,(S)) defined by the formula <41) <12:/><:> vs). e > so. otherwise. ll ll Q *1 By Corollary 22, it follows that there is ...

Page 1669

Let .11 :1, —>I, be a

I, whenever C is a compact subset of I,; (b) (M(-)),eC°°(I,), j= l,....n,. Then (i) for

each (p in C°°(I,), <poM will denote the function Ip in C°°(I1) defined, for .1: in I1, ...

Let .11 :1, —>I, be a

**mapping**of I1 into I, such that (a) M*'C is a compact subset ofI, whenever C is a compact subset of I,; (b) (M(-)),eC°°(I,), j= l,....n,. Then (i) for

each (p in C°°(I,), <poM will denote the function Ip in C°°(I1) defined, for .1: in I1, ...

Page 1736

The

Lemmas 3.22 and 3.23, and evidently

Definition 3.15 that it

H§ ...

The

**mapping**g —> Eg[C is a continuous**mapping**of H"')(e'1I) into H“"(C) byLemmas 3.22 and 3.23, and evidently

**maps**C§°(I) into C§°(C). It follows fromDefinition 3.15 that it

**maps**H{,'l(s'1I) into Hf,'l(C). Thus, f£=E(foS;1)lC belongs toH§ ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero