## Linear Operators, Part 2 |

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Results 1-3 of 83

Page 918

and since n g pi, it follows that ;1(e,,+1—é',_+1) = 0 and ,u(€,,+1——e,,+[) = O.

Suppose that ;4(e,,+1—€,,+1) > 0 and let 0' be an

en +1—é',, +1 such that 0 < /4(0) < co. Let the vectors /1, . . ., f"+1 in St be defined

by ...

and since n g pi, it follows that ;1(e,,+1—é',_+1) = 0 and ,u(€,,+1——e,,+[) = O.

Suppose that ;4(e,,+1—€,,+1) > 0 and let 0' be an

**arbitrary**fixed Borel subset ofen +1—é',, +1 such that 0 < /4(0) < co. Let the vectors /1, . . ., f"+1 in St be defined

by ...

Page 1179

Theorem 25 remaim valid in the /ull range 1 < p < oo, and for junctions f with

values in an

with trivial modifications of its proof to functions with values in an

space ...

Theorem 25 remaim valid in the /ull range 1 < p < oo, and for junctions f with

values in an

**arbitrary**Hilbert spare. PROOF. Note that Theorem 20 goes overwith trivial modifications of its proof to functions with values in an

**arbitrary**Hilbertspace ...

Page 1337

Q.E.D. \Ve have seen in Theorem 1 and Corollary 2 that an

(I) has an expansion of "Fourier integral” type in terms of eigenfunctions W,(t, 1.)

of the differential operator 1'. Unfortunately, the interest of Theorem l is more ...

Q.E.D. \Ve have seen in Theorem 1 and Corollary 2 that an

**arbitrary**vector f in L2(I) has an expansion of "Fourier integral” type in terms of eigenfunctions W,(t, 1.)

of the differential operator 1'. Unfortunately, the interest of Theorem l is more ...

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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