## Linear Operators, Part 2 |

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Page 1223

How are we to choose its

the collection fl), of all functions with one continuous derivative. If f and g are any

two such functions, we have om. g) = I; 1"/'u>gT>d¢ = J;/<¢>o'<oo+¢</<1> ...

How are we to choose its

**domain**? A natural first guess is to choose as**domain**the collection fl), of all functions with one continuous derivative. If f and g are any

two such functions, we have om. g) = I; 1"/'u>gT>d¢ = J;/<¢>o'<oo+¢</<1> ...

Page 1249

P2112, the final

is a projection if P is a partial isometry. Let .11, v EEIR, the initial

Then the identity |z+vl2 = |P.r+Pv[2 shows that (x, v)+(v, ac) = (Pm, Pv)+(Pv, Pw).

P2112, the final

**domain**of P. To complete the proof it will suffice to show that P*Pis a projection if P is a partial isometry. Let .11, v EEIR, the initial

**domain**of P.Then the identity |z+vl2 = |P.r+Pv[2 shows that (x, v)+(v, ac) = (Pm, Pv)+(Pv, Pw).

Page 1669

Let I, hc a

of I1 into I, such that (a) M*'C is a compact subset of 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 ...

Let I, hc a

**domain**in E", and let I, be a**domain**in E". Let .11 :1, —>I, be a mappingof I1 into I, such that (a) M*'C is a compact subset of 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 ...

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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