## Linear Operators, Part 2 |

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Page 1174

is in none of the sets ej. Then <48) I go)/<s>d.~ =

from the similar well-known equation for scalarvalued functions.

**Let**h be a scalar-valued function and put g(s) = h(s)a:':' if s is in e, and g(s) = 0 if sis in none of the sets ej. Then <48) I go)/<s>d.~ =

**f**ho)!/(->|<a; hence (42) followsfrom the similar well-known equation for scalarvalued functions.

Page 1656

23 LEMMA.

an integer. Then (i) the correspondence

into itself and 0/ H"')(I) into itself; (ii) the correspondence

23 LEMMA.

**Let**I be an open set in E".**Let**Io be an open subset 0/ I and**let**k bean integer. Then (i) the correspondence

**F**—>**F**is a continuous mapping of A“”(I)into itself and 0/ H"')(I) into itself; (ii) the correspondence

**F**->**F**{I° is a continuous ...Page 1696

Hence, we can evidently suppose without loss of generality that the carrier C of F

is a compact subset of I. If I is ...

integer, and let the carrier C of F be a compact subset of I. Then F is in Hg'l(I).

PRooF ...

Hence, we can evidently suppose without loss of generality that the carrier C of F

is a compact subset of I. If I is ...

**Let F**be in H"'l(I), where k is a non-negativeinteger, and let the carrier C of F be a compact subset of I. Then F is in Hg'l(I).

PRooF ...

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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