## Linear Operators, Part 2 |

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Page 1247

Q.E.D. Next we shall require some information on positive self adjoint

transformations and their square roots. 2 LEMMA. A self adjoint transformation T

is positive if and only if a(T) is a subset of the interval [0, oo).

Q.E.D. Next we shall require some information on positive self adjoint

transformations and their square roots. 2 LEMMA. A self adjoint transformation T

is positive if and only if a(T) is a subset of the interval [0, oo).

**Pnoor**. Let E be the ...Page 1474

zero between every pair of zeros of a(t, 1.1), we have only to show that the

interval (a, z] between a and the smallest zero z of a(t, /ll) contains a zero of a(t, 1.

), and we ...

**Pnoor**. Suppose this is false. Then 1.l < 1.. Since, by Lemma 35. a(t, 1.) has azero between every pair of zeros of a(t, 1.1), we have only to show that the

interval (a, z] between a and the smallest zero z of a(t, /ll) contains a zero of a(t, 1.

), and we ...

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1-¢)(w)(l, /11)}d¢ )a(z1,1,)a'(z1, 11) < 0. This contradiction establishes our

assertion. Q.E.D. O nu/\ “fir. 1: -» " I 4-6 COROLLARY. The kth zero of a(t, 1) (the

zeros being counted in ascending order) is a monotone decreasing /unction 0/ 1.

1-¢)(w)(l, /11)}d¢ )a(z1,1,)a'(z1, 11) < 0. This contradiction establishes our

assertion. Q.E.D. O nu/\ “fir. 1: -» " I 4-6 COROLLARY. The kth zero of a(t, 1) (the

zeros being counted in ascending order) is a monotone decreasing /unction 0/ 1.

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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