## Linear Operators: Spectral operators |

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Page 1015

If lim T. = T in the

the integral in [+] contains o (T,) for all sufficiently large n. From Corollary VII.6.3 it

is seen that, in the

If lim T. = T in the

**norm**of HS it follows from Lemma VII.6.5 that the contour C ofthe integral in [+] contains o (T,) for all sufficiently large n. From Corollary VII.6.3 it

is seen that, in the

**norm**of HS1, lim [A, -T.]−1 = [A, -T]−1 *—-co uniformly for A in ...Page 1297

The first

Now Ti(t) is an adjoint (Theorem 10); therefore (cf. XII.1.6) T(T(r)) is complete in

the

of ...

The first

**norm**is the**norm**of the pair [f, Tif) as an element of the graph of Ti(r).Now Ti(t) is an adjoint (Theorem 10); therefore (cf. XII.1.6) T(T(r)) is complete in

the

**norm**|f||1. Since the two additional terms in fle are the**norm**of f as an elementof ...

Page 1782

... a

? is a normed linear space. The

for example, any one of the following

... a

**norm**||, i.e., if each of the spaces 3., is a normed linear space, then the space? is a normed linear space. The

**norm**in 3 may be introduced in a variety of ways;for example, any one of the following

**norms**defines the product topology in 3.### What people are saying - Write a review

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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