## Linear Operators: Spectral theory |

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Page 1447

Then (a) if n is odd, ae(r) is the entire

0, ae(r) is the positive half- axis bounded below by n Aq = min ^ ViWY- — oo<(<+

oo j=0 Proof. Since r is formally self adjoint, (rf, f) = (/, t/) for / in S)(T0(t)), so that ...

Then (a) if n is odd, ae(r) is the entire

**real axis**; (b) if n is even, and (— l)n/2qn 2?0, ae(r) is the positive half- axis bounded below by n Aq = min ^ ViWY- — oo<(<+

oo j=0 Proof. Since r is formally self adjoint, (rf, f) = (/, t/) for / in S)(T0(t)), so that ...

Page 1550

E7 Suppose that for some (real or complex) X every solution of the equation (X—

r)f = 0 is of class Lp(I) and every ... Prove that the essential spectrum of the

operator 7\(t, p) is the positive semi-axis if n is even, and the entire

odd.

E7 Suppose that for some (real or complex) X every solution of the equation (X—

r)f = 0 is of class Lp(I) and every ... Prove that the essential spectrum of the

operator 7\(t, p) is the positive semi-axis if n is even, and the entire

**real axis**if n isodd.

Page 1597

Then the essential spectrum of t in the interval [0, oo) is the semi-axis [c, oo) (

Naimark [5])- (21 ) Suppose that q tends monotonically to — oo in the interval [0,

oo ) and that q(t) = o(fi). Then the essential spectrum of t is the entire

Then the essential spectrum of t in the interval [0, oo) is the semi-axis [c, oo) (

Naimark [5])- (21 ) Suppose that q tends monotonically to — oo in the interval [0,

oo ) and that q(t) = o(fi). Then the essential spectrum of t is the entire

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive 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