## Linear Operators: Spectral operators |

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Page 1000

A similar argument shows that f,(z) → f, (3)

the half-plane Z(z) > 0. If {fi} were known to be

neighborhood of U, the analyticity of its limit fo would be clear. Unfortunately it is

not clear ...

A similar argument shows that f,(z) → f, (3)

**uniformly**on each compact subset onthe half-plane Z(z) > 0. If {fi} were known to be

**uniformly**convergent in aneighborhood of U, the analyticity of its limit fo would be clear. Unfortunately it is

not clear ...

Page 1001

It is clear that f, converges

neither a nor b. Let M be a bound for the sequence p, so that • M f,(a+ is) is Ms |e-

to le"dr = — , — 1 < s - 0. 0 |s| In the same way it is seen that |f,(a+is) < Ms- when

...

It is clear that f, converges

**uniformly**on any portion of Q whose closure containsneither a nor b. Let M be a bound for the sequence p, so that • M f,(a+ is) is Ms |e-

to le"dr = — , — 1 < s - 0. 0 |s| In the same way it is seen that |f,(a+is) < Ms- when

...

Page 1108

If 2,0m) is a parameter family of sequences such that (i) A, (m) → A, as m -- Co

is bounded

...

If 2,0m) is a parameter family of sequences such that (i) A, (m) → A, as m -- Co

**uniformly**in i, (ii) lim, ... X., |2,(m)}* = 0**uniformly**in m, it follows that (iii) X., |2,(m)*is bounded

**uniformly**in m, (iv) A, (m) → 0 as i → CO**uniformly**in m. Thus, by the...

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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