## Linear Operators: Spectral theory |

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Page 1087

Let Tp be a 1 -parameter family of bounded operators defined in a sub- interval /

of the parameter interval 1 5S p g oo, each operator Tv acting in the space LV(S,

E, /u).

Let Tp be a 1 -parameter family of bounded operators defined in a sub- interval /

of the parameter interval 1 5S p g oo, each operator Tv acting in the space LV(S,

E, /u).

**Suppose**that for px, p2 in /, T„ and Tv always agree on the intersection of ...Page 1452

Proof.

dx)x\* ^ fi\x\2, x e ®(T), so that if \x\ is bounded, (Tx, x) is bounded below.

Conversely,

Theorem XII.

Proof.

**Suppose**that such a /j, exists. Then, by Theorem XII. 2. 6, (Tx, x) = f " X\E(dx)x\* ^ fi\x\2, x e ®(T), so that if \x\ is bounded, (Tx, x) is bounded below.

Conversely,

**suppose**that for each n, en = ( — oo, — n) r\ a(T) is non-void. ByTheorem XII.

Page 1597

(18) In the interval [0, oo),

, <~o° \q(t)\3 c )* v ' dt < 00, for large M. Then the essential spectrum of r is empty (

Wintner [8] ). (19) In the interval [a, co),

(18) In the interval [0, oo),

**suppose**that (a) lim q(t) = — oo, (-.CO (b) lim sup \ < co, <~o° \q(t)\3 c )* v ' dt < 00, for large M. Then the essential spectrum of r is empty (

Wintner [8] ). (19) In the interval [a, co),

**suppose**that lim q(t) = c. (-i> Then the ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

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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 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