## Linear Operators: Spectral theory |

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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 an of t, we can write the equation (t— A)/ = 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 an of t, we can write the equation (t— A)/ = 0 in the form ...

Page 1474

Then /j < A. Since, by Lemma 35, a(t, X) has a

a(t, Xt), we have only to show that the interval (a, z] between a and the smallest

Then /j < A. Since, by Lemma 35, a(t, X) has a

**zero**between every pair of**zeros**ofa(t, Xt), we have only to show that the interval (a, z] between a and the smallest

**zero**z of a(t, Aj) contains a**zero**of a(t, A), and we will have established that a(t, ...Page 1475

If we can show that <t(-, A2) has a

established that a(-, A2) has at least ra+1

A2 is in Jn. It is sufficient to prove that a(-, A^ has a

If we can show that <t(-, A2) has a

**zero**in (a, Zj] and a**zero**in [Z|, fc), we will haveestablished that a(-, A2) has at least ra+1

**zeros**in (a, 6), contradicting the fact thatA2 is in Jn. It is sufficient to prove that a(-, A^ has a

**zero**in (a, Zj], for then it will ...### What people are saying - Write a review

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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