## Linear Operators: Spectral theory |

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Page 1464

Then (a) if lim sup, ... tog(t) < —(1/4), every

number of zeros on [a, oo); (b) if lim inf, .<!”q(t) > —(1/4), no

identically zero, of ts = 0 has more than a finite number of zeros on [a, oo). PRoof.

According to ...

Then (a) if lim sup, ... tog(t) < —(1/4), every

**solution**of ts = 0 has an infinitenumber of zeros on [a, oo); (b) if lim inf, .<!”q(t) > —(1/4), no

**solution**, notidentically zero, of ts = 0 has more than a finite number of zeros on [a, oo). PRoof.

According to ...

Page 1521

Putting yo = 1/2+i so that Žo = 1 +i, we see that the equation (L1–Åo)f has one

ob. The

has ...

Putting yo = 1/2+i so that Žo = 1 +i, we see that the equation (L1–Åo)f has one

**solution**of the order of to "To as t → 00 and another which behaves like to as t →ob. The

**solution**at 20 = 1–i is exactly similar. Thus, by Theorem XII.4.19, L1–2has ...

Page 1556

What is the relationship between 0(t) and the number of zeros of a

above equation? G14 Use the result of the preceding exercise to show that if the

operator t has two boundary values at infinity, then N(t) lim = OO, t—-oo 2 where ...

What is the relationship between 0(t) and the number of zeros of a

**solution**of theabove equation? G14 Use the result of the preceding exercise to show that if the

operator t has two boundary values at infinity, then N(t) lim = OO, t—-oo 2 where ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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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