## Linear Operators: Spectral operators |

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Page 1207

It is clear from Zorn's lemma that there is a maximal set A in

spaces S3, a e A, are orthogonal. Thus to prove the ... Thus, for any finite set at

next be ...

It is clear from Zorn's lemma that there is a maximal set A in

**S**) for which thespaces S3, a e A, are orthogonal. Thus to prove the ... Thus, for any finite set at

**CA**, we have X., F(T)r,” < |F(T)rs”, a so(T), which shows that X, F(T)t,” < co. It willnext be ...

Page 1255

Thus the family {U(t)}, -oo - t < 00, is a group of unitary operators on S). It will be

shown next that the group {U(•)} is strongly continuous. Let w = f--9so be in 3,

where f is in QI. Then |U(t)+-U(t.)al” = |U(|t, —tal).

, ...

Thus the family {U(t)}, -oo - t < 00, is a group of unitary operators on S). It will be

shown next that the group {U(•)} is strongly continuous. Let w = f--9so be in 3,

where f is in QI. Then |U(t)+-U(t.)al” = |U(|t, —tal).

**c-a**'**s**” = |U(|t, -t, )rs”--|a|*–2%(U(st, ...

Page 1487

If A e o (S_), so that A is an eigenvalue of S_, there exists a non-zero fe 2 such

that S_f = 2f. Consequently, Sf = Af, so that A is an eigenvalue of S, and thus à e g

(S). This shows that q(S.)

...

If A e o (S_), so that A is an eigenvalue of S_, there exists a non-zero fe 2 such

that S_f = 2f. Consequently, Sf = Af, so that A is an eigenvalue of S, and thus à e g

(S). This shows that q(S.)

**Ca**(**S**); and similarly a (S1) C o(S), so that q(S_) U or(SI)...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

52 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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