## Linear Operators, Part 2 |

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Page 1358

a,-(s, i.)a,(t, }.)p,-,(d}.). t, -1 0 It

]* g M(J), te J, and that equation [T] holds for all f in L2(I). Q.E.D. 15 COROLLARY.

Let T, A, and {p1-1} be defined as in Theorem 14. The complement of a(T) in /1 ...

a,-(s, i.)a,(t, }.)p,-,(d}.). t, -1 0 It

**follows from Theorem**IV.8.1 that |:J'I|K(a; 1, s)|=a.9]* g M(J), te J, and that equation [T] holds for all f in L2(I). Q.E.D. 15 COROLLARY.

Let T, A, and {p1-1} be defined as in Theorem 14. The complement of a(T) in /1 ...

Page 1379

_ _ _—~3i7 7 7 {fiu} is the matrix measure of

uniquely determined for each e Q N. Since A is the union of a sequence of

neighborhoods of the same type as N, the uniqueness of {fin}

.

_ _ _—~3i7 7 7 {fiu} is the matrix measure of

**Theorem**28, the values fi“(e) areuniquely determined for each e Q N. Since A is the union of a sequence of

neighborhoods of the same type as N, the uniqueness of {fin}

**follows**immediately.

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

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