## Linear Operators: Spectral operators |

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Page 1550

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

that every solution of the equation roof = 0 is of class L.,(I) (p +q_1 = 1).

for ...

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

that every solution of the equation roof = 0 is of class L.,(I) (p +q_1 = 1).

**Prove**thatfor ...

Page 1557

Suppose that q is bounded below, and suppose that A does not belong to the

essential spectrum of t. Let f be a square-integrable solution of the equation (A–t)f

= 0, ...

**Prove**that the point A belongs to the essential spectrum of t. G20 (Wintner).Suppose that q is bounded below, and suppose that A does not belong to the

essential spectrum of t. Let f be a square-integrable solution of the equation (A–t)f

= 0, ...

Page 1568

if s, 14(1)(it = 1. H13 Suppose that [... (1+t)|q(t)|dt « Co.

the continuous spectrum of every self adjoint extension of the operator To(r).

**Prove**that a self adjoint extension of the operator has a negative eigenvalue onlyif s, 14(1)(it = 1. H13 Suppose that [... (1+t)|q(t)|dt « Co.

**Prove**that the origin lies inthe continuous spectrum of every self adjoint extension of the operator To(r).

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero