## Linear Operators: Spectral operators |

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Page 1310

Then the boundary conditions are real, and there is eractly one solution p(t, 2) of (

t–%)p = 0

exactly one solution p(t, 2) of (1–2) p = 0

Then the boundary conditions are real, and there is eractly one solution p(t, 2) of (

t–%)p = 0

**square**-**integrable**at a and satisfying the boundary conditions at a, andexactly one solution p(t, 2) of (1–2) p = 0

**square**-**integrable**at b and satisfying ...Page 1556

G16 (Hartman) Suppose that the equation ts = 0 has a solution with a finite

number of zeros. Prove that there exists a solution g of the same equation such

that g(t)- is

origin.

G16 (Hartman) Suppose that the equation ts = 0 has a solution with a finite

number of zeros. Prove that there exists a solution g of the same equation such

that g(t)- is

**square**-**integrable**on a semi-axis sufficiently far removed from theorigin.

Page 1557

lost. wo: o too so | ho o o go so d: * o o so is: st o so wo m! o (A–t)f = 0 has a

solution which is not

Prove that the point A belongs to the essential spectrum of t. G20 (Wintner).

Suppose ...

lost. wo: o too so | ho o o go so d: * o o so is: st o so wo m! o (A–t)f = 0 has a

solution which is not

**square**-**integrable**but has a**square**-**integrable**derivative.Prove that the point A belongs to the essential spectrum of t. G20 (Wintner).

Suppose ...

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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