## Linear Operators: Spectral operators |

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of T and otherwise let E(6) be the sum of all the projections E(2,) for which 2, e3,

then the function E is a resolution of the identity for T and the operational calculus

is given by the

of T and otherwise let E(6) be the sum of all the projections E(2,) for which 2, e3,

then the function E is a resolution of the identity for T and the operational calculus

is given by the

**formula**(vi) f(T) = [.., f(z)E(i). where the integral is defined as the ...Page 1089

The basic properties of the characteristic numbers u,(T) are stated in the following

lemma and corollaries. 2 LEMMA. The characteristic numbers u,(T) of a compact

operator are given by the following

The basic properties of the characteristic numbers u,(T) are stated in the following

lemma and corollaries. 2 LEMMA. The characteristic numbers u,(T) of a compact

operator are given by the following

**formula**: plari (T) = min Inax |Tops, n > 0.Page 1363

so basis for this

projection in the resolution of the identity for T corresponding to (A1, A2) may be

calculated from the resolvent by the

ie; ...

so basis for this

**formula**is found in Theorem XII.2.10 which asserts that theprojection in the resolution of the identity for T corresponding to (A1, A2) may be

calculated from the resolvent by the

**formula**1 FA, -o E((A1, A,))f = lim lim - [R(A—ie; ...

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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