## Linear Operators: Spectral operators |

### From inside the book

Results 1-3 of 71

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 1307

D, are

)1),(g)–D,(f)01(g), f, geo(T(r)). PRoof. Let A be any

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

**boundary values**at b, such that (rf, g)–(f, Tg) = C, (f)C2(g)–C2(f)C1(g) +D.(f)1),(g)–D,(f)01(g), f, geo(T(r)). PRoof. Let A be any

**boundary value**for t. Since t ...Page 1471

if r has no

two real

set of boundary conditions defining a self adjoint restriction T of Ti(t) is of the form

...

if r has no

**boundary values**at b; while if t has**boundary values**at b, we may findtwo real

**boundary values**D1, D, for T., ... By Theorem 2.30 and Corollary 2.31, aset of boundary conditions defining a self adjoint restriction T of Ti(t) is of the form

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

52 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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