## Linear Operators: Spectral operators |

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Page 1037

product clearly converges to zero for A = A, it is readily seen that the function pa(T

) is

0, then pa(T) is continuous in T relative to the Hilbert-Schmidt norm in HS.

product clearly converges to zero for A = A, it is readily seen that the function pa(T

) is

**analytic**for A # 0 and vanishes only for A in g(T). It remains to show that if A #0, then pa(T) is continuous in T relative to the Hilbert-Schmidt norm in HS.

Page 1040

yı (A) is

)*y vanishes which will prove that y(?) is

(A) can only fail to be

yı (A) is

**analytic**even at A = A,. It will now be shown that y,(A) = A*E(7.3 T)*R(Å; T)*y vanishes which will prove that y(?) is

**analytic**at all the points A = %, , so that y(A) can only fail to be

**analytic**at the point A = 0. To show this, note that (u,(2), ...Page 1102

The determinant det(1+2T,) is an

T, operates in finite-dimensional space, and hence more generally if T, has a

finite-dimensional range. Thus, since a bounded convergent sequence of

The determinant det(1+2T,) is an

**analytic**(and even a polynomial) function of z, ifT, operates in finite-dimensional space, and hence more generally if T, has a

finite-dimensional range. Thus, since a bounded convergent sequence of

**analytic**...### 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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### 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