## Linear Operators: Spectral theory |

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Page 1358

where K(G;t,s) = J f GWoi{s,X)aj(t,X)Pij(dX). i, 3=1 J e It

8.1 that [j^G;*,*)!^]* <M(J), teJ, and that equation [t] holds for all / in L2(I). Q.E.D.

15 Corollary. Let T, A, and {p^} be defined as in Theorem 14. The complement of

...

where K(G;t,s) = J f GWoi{s,X)aj(t,X)Pij(dX). i, 3=1 J e It

**follows from Theorem**IV.8.1 that [j^G;*,*)!^]* <M(J), teJ, and that equation [t] holds for all / in L2(I). Q.E.D.

15 Corollary. Let T, A, and {p^} be defined as in Theorem 14. The complement of

...

Page 1379

is the matrix measure of

for each e QN. Since A is the union of a sequence of neighborhoods of the same

type as N, the uniqueness of {pif}

is the matrix measure of

**Theorem**23, the values pu(e) are uniquely determinedfor each e QN. Since A is the union of a sequence of neighborhoods of the same

type as N, the uniqueness of {pif}

**follows**immediately. Q.E.D. 27**Theorem**. Let r ...Page 1400

Taking together Lemmas 7, 9, and Corollary 8, we obtain the

which shows the extent to which the spectrum of a self adjoint operator derived

from a formal differential operator depends on the boundary conditions involved.

Taking together Lemmas 7, 9, and Corollary 8, we obtain the

**following theorem**,which shows the extent to which the spectrum of a self adjoint operator derived

from a formal differential operator depends on the boundary conditions involved.

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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