## Linear Operators: Spectral operators |

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Page 1088

[s, t) in SXS, and suppose that K(s, t) = K(t, s), so that (Ki)(a) = | K(s,t)f(t)h(d)

defines a

eigenfunctions of K, and {u,} an enumeration of the corresponding eigenvalues.

Show that ...

[s, t) in SXS, and suppose that K(s, t) = K(t, s), so that (Ki)(a) = | K(s,t)f(t)h(d)

defines a

**compact operator**in L2(S, 2, 11). Let {q,} be an enumeration of theeigenfunctions of K, and {u,} an enumeration of the corresponding eigenvalues.

Show that ...

Page 1089

These numbers are called the characteristic numbers of the operator T; we write

u,(T) for the nth characteristic number of T. In terms of these characteristic

numbers, we may define various norms for and classes of

These numbers are called the characteristic numbers of the operator T; we write

u,(T) for the nth characteristic number of T. In terms of these characteristic

numbers, we may define various norms for and classes of

**compact operators**.Page 1095

If T, e C, is a sequence of operators such that |T,-T,), → 0 as m, n -> 00, there

eacists a

PRoof. By Lemma 9(a) and the fact that the family of

in ...

If T, e C, is a sequence of operators such that |T,-T,), → 0 as m, n -> 00, there

eacists a

**compact operator**T such that T. -> T (in the topology of C,) as n => oc.PRoof. By Lemma 9(a) and the fact that the family of

**compact operators**is closedin ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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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