## Linear Operators: Spectral theory |

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Page 1226

Part (a)

from part (a) and Lemma 5(c). Q.E.D. It follows from Lemma 6(b) that any

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Part (a)

**follows immediately**from Lemma 5(b), and part (b)**follows immediately**from part (a) and Lemma 5(c). Q.E.D. It follows from Lemma 6(b) that any

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Page 1469

It

–e/2)|f|* for f in Q(T.). Since U, e.,3 (T.) D o(To(t)), ((t–(Äo-e/2))f, f) > 0 for f in Q(To(t

)), so that t—(Åo-e/2) is formally positive, and, by Corollary 30, t is finite below ...

It

**follows immediately**from Theorems 4.1, 4.2, and XII.7.2 that (rf, f) = (T. f. f) > (20–e/2)|f|* for f in Q(T.). Since U, e.,3 (T.) D o(To(t)), ((t–(Äo-e/2))f, f) > 0 for f in Q(To(t

)), so that t—(Åo-e/2) is formally positive, and, by Corollary 30, t is finite below ...

Page 1478

Thus A, -> 00 by Corollary 26 and Corollary 27. The uniqueness of q,

distinct eigenvalues have different numbers of zeros, and since by Corollary 44

this ...

Thus A, -> 00 by Corollary 26 and Corollary 27. The uniqueness of q,

**follows****immediately**from Lemma 41. Since, by Lemma 45, eigenfunctions belonging todistinct eigenvalues have different numbers of zeros, and since by Corollary 44

this ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete 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 identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero