## Linear Operators: Spectral operators |

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Page 1358

where K(G; t, s) = X G(A)0,0s, A)o,(t, A)p,(dž). i, j=1 J e It

8. I that [s, K.G. . )*dj's M(j), tes, and that equation [*] holds for all f in L2(I). Q.E.D.

15 CoRollARY. Let T, A, and {p,} be defined as in Theorem 14. The complement ...

where K(G; t, s) = X G(A)0,0s, A)o,(t, A)p,(dž). i, j=1 J e It

**follows from Theorem**IV.8. I that [s, K.G. . )*dj's M(j), tes, and that equation [*] holds for all f in L2(I). Q.E.D.

15 CoRollARY. Let T, A, and {p,} be defined as in Theorem 14. The complement ...

Page 1379

to o so no {5,) is the matrix measure of

determined for each e C N. Since A is the union of a sequence of neighborhoods

of the same type as N, the uniqueness of {6,3

to o so no {5,) is the matrix measure of

**Theorem**23, the values 6,06) are uniquelydetermined for each e C N. Since A is the union of a sequence of neighborhoods

of the same type as N, the uniqueness of {6,3

**follows**immediately. Q.E.D. 27 ...Page 1381

Since [0,1] is a closed interval, it

spectrum of To consists entirely of isolated points, every such point being an

eigenvalue of Ta, and that To has a complete set of orthonormal eigenfunctions.

Since [0,1] is a closed interval, it

**follows from Theorems**4.1 and 4.2 that thespectrum of To consists entirely of isolated points, every such point being an

eigenvalue of Ta, and that To has a complete set of orthonormal eigenfunctions.

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### 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