## Linear Operators: Spectral operators |

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Page 861

Since |a| = |re1 = |T.e| < |es|T, it follows that the

see that t is also continuous it will first be shown that r(3) is closed in B(3). To do

this the following criterion is useful: an element Te B(3) is in r(3) if and only if (Ty)

2 ...

Since |a| = |re1 = |T.e| < |es|T, it follows that the

**inverse**map T' is continuous. Tosee that t is also continuous it will first be shown that r(3) is closed in B(3). To do

this the following criterion is useful: an element Te B(3) is in r(3) if and only if (Ty)

2 ...

Page 877

Then an element y in Q) has an

Consequently the spectrum of y as an element of Q) is the same as its spectrum

as an element of 3. A o Proof. If yol exists as an element of Q) then, since 3 and 9)

...

Then an element y in Q) has an

**inverse**in 3: if and only if it has an**inverse**in Q).Consequently the spectrum of y as an element of Q) is the same as its spectrum

as an element of 3. A o Proof. If yol exists as an element of Q) then, since 3 and 9)

...

Page 1311

It will be convenient in stating many of the results to suppose that the number A =

0 is in p(T), that is, that T has a bounded everywhere defined

convenient assumption is equivalent to the supposition that the operator r has

been ...

It will be convenient in stating many of the results to suppose that the number A =

0 is in p(T), that is, that T has a bounded everywhere defined

**inverse**. Thisconvenient assumption is equivalent to the supposition that the operator r has

been ...

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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