## Linear Operators, Part 2 |

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Page 1550

Bellman) Suppose that every solution of the equation If = 0 is of class L,(I) and

that every solution of the equation 1*] = 0 is of class La(I) (p'1+q“1 = 1).

for ...

**Prove**that the essential spectrum of the operator r in L,(I) is the empty set. E8 (Bellman) Suppose that every solution of the equation If = 0 is of class L,(I) and

that every solution of the equation 1*] = 0 is of class La(I) (p'1+q“1 = 1).

**Prove**thatfor ...

Page 1554

Write g(t) = h(t)f(t) and obtain, by variation of parameters, an equation of type (A)

as in G7 for h. Infer that 8(1) = 0(f(l))-l G9 Let N (t) be the number of zeros in the ...

**Prove**that the equation 8"(i)—qz(i)8(i) = 0 has a square-integrable solution. (Hint:Write g(t) = h(t)f(t) and obtain, by variation of parameters, an equation of type (A)

as in G7 for h. Infer that 8(1) = 0(f(l))-l G9 Let N (t) be the number of zeros in the ...

Page 1563

contains the positive semi-axis. (Hint: Apply Theorem 7.1.) G41 Suppose that the

function q is bounded below. Suppose that the origin belongs to the essential ...

**Prove**that 10-1)/.1 = 0(~/<b.—a.>)(b)**Prove**that the essential spectrum of 1contains the positive semi-axis. (Hint: Apply Theorem 7.1.) G41 Suppose that the

function q is bounded below. Suppose that the origin belongs to the essential ...

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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