## Linear Operators: Spectral theory |

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Page 1297

Q.E.D. We now turn to a discussion of the specific form assumed in the present

case by the abstract “

see that the discussion leads us to a number of results about deficiency indices.

Q.E.D. We now turn to a discussion of the specific form assumed in the present

case by the abstract “

**boundary values**” introduced in the last chapter. We shallsee that the discussion leads us to a number of results about deficiency indices.

Page 1298

If t is formally self adjoint, then Definition 17 of a

with Definition XII.4.20 of a

theorem gives a basic property of

the ...

If t is formally self adjoint, then Definition 17 of a

**boundary value**for t coincideswith Definition XII.4.20 of a

**boundary value**for To(t). PRoof. ... Q.E.D. The nexttheorem gives a basic property of

**boundary values**of differential operators. Forthe ...

Page 1307

D2 are

)ID2(g)–D2(f) D1(g), f, ge 3 (Ti(r)). PRoof. Let A be any

**boundary values**C1, C2, D1, D, where C1, C2 are**boundary values**at a and D1,D2 are

**boundary values**at b, such that (tf, g)–(f, ty) = C, (f)C2(g)–C2(f)C1(g) +D1(f)ID2(g)–D2(f) D1(g), f, ge 3 (Ti(r)). PRoof. Let A be any

**boundary value**for t.### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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### Common terms and phrases

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