## Linear Operators, Part 2 |

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Page 992

It follows from Lemma 3.6(i) that ,u(V) is finite and thus, as was observed in the

note following the proof of that lemma, for every positive ... Since fin vanishes on

the complement of V it follows from the

It follows from Lemma 3.6(i) that ,u(V) is finite and thus, as was observed in the

note following the proof of that lemma, for every positive ... Since fin vanishes on

the complement of V it follows from the

**preceding lemma**that Q5/n does likewise.Page 1192

Q.E.D. 2 LEMMA. The spectrum of a self adjoint operator T is real and the

resolvent is a normal operator with R(ai; T)' = R(6i; T) and lR(<1; T)! § l~»"(¢)|“. -/(

1) ¢ 0PROOF. Let an be a non-real scalar. The

—T)'1 ...

Q.E.D. 2 LEMMA. The spectrum of a self adjoint operator T is real and the

resolvent is a normal operator with R(ai; T)' = R(6i; T) and lR(<1; T)! § l~»"(¢)|“. -/(

1) ¢ 0PROOF. Let an be a non-real scalar. The

**preceding lemma**shows that (oiI—T)'1 ...

Page 1474

We will see below what significance this alternative has for the spectrum of T. 4-3

LEMMA. If 1. is in J,, and ll is in J,,+l, then 1. ... By the remark (a) made

We will see below what significance this alternative has for the spectrum of T. 4-3

LEMMA. If 1. is in J,, and ll is in J,,+l, then 1. ... By the remark (a) made

**preceding****Lemma**4-1, 0 < (A-/1,)f a(t, /mo, .1,)a - jj {<w><:. amt.1.)-a<¢.1.><w><¢.a>}d1 ...### What people are saying - Write a review

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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