## Linear Operators: Spectral theory |

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Page 1290

=o is

same way, the formal differential operator (il2)(d[dt)n{p{t)(d[dt) + {dldt)p(t)}(d/dt)n

is

=o is

**formally self adjoint**provided only that the coefficients pi are real. In thesame way, the formal differential operator (il2)(d[dt)n{p{t)(d[dt) + {dldt)p(t)}(d/dt)n

is

**formally self adjoint**provided that p(t) is a real function. If we use these ...Page 1295

the (regular or irregular) formal differential operator x is

the operator T0(r) is symmetric. Proof. Clearly T0(r) Q Tx(x). Corollary 5 shows

that Tx(r) Q T0(r)*. Q.E.D. We recall (cf. Definition XII. 4. 9 ) that if t is formally self ...

the (regular or irregular) formal differential operator x is

**formally self adjoint**thenthe operator T0(r) is symmetric. Proof. Clearly T0(r) Q Tx(x). Corollary 5 shows

that Tx(r) Q T0(r)*. Q.E.D. We recall (cf. Definition XII. 4. 9 ) that if t is formally self ...

Page 1400

Let x be a

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

the essential spectrum of x. Then the deficiency indices of x are both equal to an

...

Let x be a

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

the essential spectrum of x. Then the deficiency indices of x are both equal to an

...

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### Contents

BAlgebras | 859 |

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 constant 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero