## Linear Operators: Spectral theory |

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Page 1087

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

hypotheses of Exercise 50 be satisfied. Show that g(T,) Co.(T,) n g(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, u). Prove that log o (T,) is a convex function of p. 51 Let the

hypotheses of Exercise 50 be satisfied. Show that g(T,) Co.(T,) n g(T,) if pi sp2 s

pa; p1, ...

Page 1563

Prove that (4–1)f, = O(V(b.-a,)). (b) Prove that the essential spectrum of r contains

the positive semi-axis. (Hint: Apply Theorem 7.1.) G41

q is bounded below.

Prove that (4–1)f, = O(V(b.-a,)). (b) Prove that the essential spectrum of r contains

the positive semi-axis. (Hint: Apply Theorem 7.1.) G41

**Suppose**that the functionq is bounded below.

**Suppose**that the origin belongs to the essential spectrum ...Page 1602

(47) In [0, oo),

solutions f and g such that |f(s)'ds = o(e) and sig'(s)ods = o(e). Then the point A

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

...

(47) In [0, oo),

**suppose**that the equation (2–1)f = 0 has two linearly independentsolutions f and g such that |f(s)'ds = o(e) and sig'(s)ods = o(e). Then the point A

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

**Suppose**...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero