## Linear Operators: Spectral theory |

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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 (2–1)

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 (2–1)

f = 0, ...

Page 1563

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 spectrum of

...

**Prove**that (4–1)f, - O(V(b.-a,)). (b)**Prove**that the essential spectrum of t containsthe 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 spectrum of

...

Page 1568

if | ta(t) di = 1. H13 Suppose that | (1+1)*(t) di < 0.

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 | ta(t) di = 1. H13 Suppose that | (1+1)*(t) di < 0.

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

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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