## Linear Operators: Spectral theory |

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Page 1436

Let {g,} be a bounded sequence of elements of £(T) such that {Tg,}

Find a subsequence {g,} = {h} such that r' (h,)

h, = h,-X pro(h)p, is in 3), and Th, – Th, . Thus (h,

Let {g,} be a bounded sequence of elements of £(T) such that {Tg,}

**converges**.Find a subsequence {g,} = {h} such that r' (h,)

**converges**for each j, 1 < j < k. Thenh, = h,-X pro(h)p, is in 3), and Th, – Th, . Thus (h,

**converges**, so that {h} = {h ...Page 1461

Let u > 0; we shall show, using Corollary 2, that —(u-He)go.(t+11). If this is false,

then by Corollary 2 there exists a bounded sequence {f,} of £(To(t+11)) such that

{(t+11+u-He)f,}

Let u > 0; we shall show, using Corollary 2, that —(u-He)go.(t+11). If this is false,

then by Corollary 2 there exists a bounded sequence {f,} of £(To(t+11)) such that

{(t+11+u-He)f,}

**converges**, but such that {fi} has no convergent subsequence.Page 1664

Then the expression F, - F(e-o") is called the Lth Fourier coefficient of F. The

formal series (2+)-" X Freit * L is called the Fourier series of F. 39 LEMMA. The

Fourier series of an element F in D, (C)

follows ...

Then the expression F, - F(e-o") is called the Lth Fourier coefficient of F. The

formal series (2+)-" X Freit * L is called the Fourier series of F. 39 LEMMA. The

Fourier series of an element F in D, (C)

**converges**unconditionally to F. PRoof. Itfollows ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero