## Linear Operators, Part 2 |

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Page 1290

4 is

way, the formal differential operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is

observations ...

4 is

**formally self adjoint**provided only that the coefficients pi are real. In the sameway, the formal differential operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is

**formally self adjoint**provided that p(t) is a real function. If we use theseobservations ...

Page 1295

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

the operator To("r) is symmetric. PROOF. Clearly To("r) Q T1(-r). Corollary 5

shows that T,(r) Q To(t)"'. Q.E.D. We recall (cf. Definition XII.4-.9) that ifr is formally

...

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

**formally self adjoint**thenthe operator To("r) is symmetric. PROOF. Clearly To("r) Q T1(-r). Corollary 5

shows that T,(r) Q To(t)"'. Q.E.D. We recall (cf. Definition XII.4-.9) that ifr is formally

...

Page 1400

Let T be a

with at least one fixed end point. Let J. be an arbitrary real point not belonging to

the essential spectrum of 1'. Then the deficiency indices of 1 are both equal to ...

Let T be a

**formally self adjoint**formal differential operator defined on an interval Iwith at least one fixed end point. Let J. be an arbitrary real point not belonging to

the essential spectrum of 1'. Then the deficiency indices of 1 are both equal to ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients 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 hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero