## Linear Operators: Spectral operators |

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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 a(T,) cost,) n g(T,) if p1 is pz

spa; p1, ...

**Suppose**that for pi, p, in I, T, and T, always agree on the intersection of L,(S, X, 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 a(T,) cost,) n g(T,) if p1 is pz

spa; p1, ...

Page 1144

15 THEOREM. Let 1 < p < 00, and let the compact operator T in Hilbert space \)

have an anti-Hermitian part lying in the class C. Let y1,..., y, be non-overlapping,

differentiable arcs in the compler plane starting at the origin.

of ...

15 THEOREM. Let 1 < p < 00, and let the compact operator T in Hilbert space \)

have an anti-Hermitian part lying in the class C. Let y1,..., y, be non-overlapping,

differentiable arcs in the compler plane starting at the origin.

**Suppose**that eachof ...

Page 1602

(47) In [0, oo),

solutions f and g such that |f(s)ods = 0(e) and [.. g'(s)*ds = 0(e) Then the point %

belongs to the essential spectrum of t (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)ods = 0(e) and [.. g'(s)*ds = 0(e) 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 |

4 Exercises | 879 |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero