## Linear Operators: Spectral theory |

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Page 1290

that since to ti - (T211)*, the operator X (–) () to () is

only that the coefficients p, are real. In the same way, the formal differential

operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is

p(t) is ...

that since to ti - (T211)*, the operator X (–) () to () is

**formally self adjoint**providedonly that the coefficients p, are real. In the same way, the formal differential

operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is

**formally self adjoint**provided thatp(t) is ...

Page 1295

If the (regular or irregular) formal differential operator t is

the operator To(r) is symmetric. PRoof. Clearly To(r) C Ti(t). Corollary 5 shows

that Ti(r) C To(t)*. Q.E.D. We recall (cf. Definition XII.4.9) that if t is formally self ...

If the (regular or irregular) formal differential operator t is

**formally self adjoint**thenthe operator To(r) is symmetric. PRoof. Clearly To(r) C Ti(t). Corollary 5 shows

that Ti(r) C To(t)*. Q.E.D. We recall (cf. Definition XII.4.9) that if t is formally self ...

Page 1464

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. be a real,

Then (a) if lim sup, ... tog(t) < —(1/4), every solution of ts = 0 has an infinite

number ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. be a real,

**formally self adjoint**formal differential operator defined on an interval [a, oo).Then (a) if lim sup, ... tog(t) < —(1/4), every solution of ts = 0 has an infinite

number ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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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