## Linear Operators: Spectral theory |

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Page 1420

Then, assuming that t+t' has a non-

3 (T1(t +t')) and 3)(Ti(r)) have the same elements and equivalent topologies; (B)

the differential operators t and t' have the same deficiency indices. PRoof.

Then, assuming that t+t' has a non-

**zero**leading coefficient, (A) the Hilbert spaces3 (T1(t +t')) and 3)(Ti(r)) have the same elements and equivalent topologies; (B)

the differential operators t and t' have the same deficiency indices. PRoof.

Page 1432

Suppose first that the end point under consideration is finite so that without loss of

generality we can suppose it to be at

the leading coefficient a, of r, we can write the equation (1–2)f = 0 in the form ...

Suppose first that the end point under consideration is finite so that without loss of

generality we can suppose it to be at

**zero**. Then, dividing through if necessary bythe leading coefficient a, of r, we can write the equation (1–2)f = 0 in the form ...

Page 1463

Since all the terms in the integral on the right are non-negative, we must have fif,

f. f. identically

that fif," is constant. Moreover, since f, and f, have only a finite number of

c, ...

Since all the terms in the integral on the right are non-negative, we must have fif,

f. f. identically

**zero**in [c, d). Thus (fif, ')' = f°(fif, f. f.) is identically**zero**in [c,d], sothat fif," is constant. Moreover, since f, and f, have only a finite number of

**zeros**in [c, ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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### Common terms and phrases

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