## Linear Operators: Spectral operators |

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Page 1529

The confluent hypergeometric

so that " = 0, ..." = 1. Thus the Stokes lines for this

negative imaginary axes. Trying solutions of the form zoo (1 + cz + ...) and e-z ...

The confluent hypergeometric

**equation**has the characteristic**equation**&”–2 = 0,so that " = 0, ..." = 1. Thus the Stokes lines for this

**equation**are the positive andnegative imaginary axes. Trying solutions of the form zoo (1 + cz + ...) and e-z ...

Page 1553

do it to o o olo no o o G3 Suppose that the operator t has the property that for

some A the derivative of every square-integrable solution of the

0 is bounded. Prove that t has no boundary values at infinity. G4 (Wintner)

Suppose ...

do it to o o olo no o o G3 Suppose that the operator t has the property that for

some A the derivative of every square-integrable solution of the

**equation**(2–1)f =0 is bounded. Prove that t has no boundary values at infinity. G4 (Wintner)

Suppose ...

Page 1556

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

above

operator r has two boundary values at infinity, then N(t) lim — = 00, t— oc t?

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

above

**equation**? G14 Use the result of the preceding exercise to show that if theoperator r has two boundary values at infinity, then N(t) lim — = 00, t— oc t?

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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