## Linear Operators: Spectral operators |

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Page 1435

o where, however, the infinite series constituting the final factor of this expression

is not convergent but is a divergent

c', co, . . . can be determined by formally substituting the

o where, however, the infinite series constituting the final factor of this expression

is not convergent but is a divergent

**asymptotic**series. The coefficients k”, ..., k”, e,c', co, . . . can be determined by formally substituting the

**asymptotic**expression ...Page 1529

Then in this angle any solution of Lj = 0 whose

the factor exp (§":*-*) is exponentially small (as |z| -- 00 in the angle) relative to

any solution whose

...

Then in this angle any solution of Lj = 0 whose

**asymptotic**expansion begins withthe factor exp (§":*-*) is exponentially small (as |z| -- 00 in the angle) relative to

any solution whose

**asymptotic**expansion begins with the factor exp (-;"z"-"). Thus...

Page 1565

Show that a similar

be differentiated. H6 Let T be the self adjoint extension of To(r) obtained by the

imposition of the boundary condition B(f) = f(0) cos 0-i-f'(0) sin 0 = 0. Taking as ...

Show that a similar

**asymptotic**estimate holds for g. Show that the estimate canbe differentiated. H6 Let T be the self adjoint extension of To(r) obtained by the

imposition of the boundary condition B(f) = f(0) cos 0-i-f'(0) sin 0 = 0. Taking as ...

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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