## Linear Operators: Spectral theory |

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Page 1087

and L,(S, X, m). Prove that log o (T,) is a convex function of p. 51 Let the

hypotheses of Exercise 50 be satisfied. Show that o(T,) G o(T,) (`) o(T,) if pi sp2 s

pa; p1, ...

**Suppose**that for pi, p, in I, T, and T, always agree on the intersection of L,(S. 2, u)and L,(S, X, m). Prove that log o (T,) is a convex function of p. 51 Let the

hypotheses of Exercise 50 be satisfied. Show that o(T,) G o(T,) (`) o(T,) if pi sp2 s

pa; p1, ...

Page 1563

G41

belongs to the essential spectrum of t. (a) Let {f} be a sequence in 3 (To(r)) such

that |f| = 1, |tf, -> 0, and such that f, vanishes in the interval [0, n). Set g,(t) = f(t) sin

tv7, ...

G41

**Suppose**that the function q is bounded below.**Suppose**that the originbelongs to the essential spectrum of t. (a) Let {f} be a sequence in 3 (To(r)) such

that |f| = 1, |tf, -> 0, and such that f, vanishes in the interval [0, n). Set g,(t) = f(t) sin

tv7, ...

Page 1602

(47) In [0, oo),

solutions f and g such that |f(s)ods = o(e) and J. g(s)"ds – ose). Then the point %

belongs to the essential spectrum of t (Hartman and Wintner [14]). (48)

(47) In [0, oo),

**suppose**that the equation (? –t)f = 0 has two linearly independentsolutions f and g such that |f(s)ods = o(e) and J. g(s)"ds – ose). Then the point %

belongs to the essential spectrum of t (Hartman and Wintner [14]). (48)

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero