## Linear Operators: Spectral theory |

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Page 876

Then (y-i-Nie)(2) = y(A) + Ni = i (1 + N), and

< y + Niel” = (y-i-Nie)(y-i-Nie)*| = |(y-i-Nie)(y–Nie)|= y?--Noel -s sy”|-|-N*. Since

this inequality must hold for all real N, a contradiction is obtained by placing N = y

” ...

Then (y-i-Nie)(2) = y(A) + Ni = i (1 + N), and

**hence**1 + NI : y + Niel.**Hence**(1+N)*< y + Niel” = (y-i-Nie)(y-i-Nie)*| = |(y-i-Nie)(y–Nie)|= y?--Noel -s sy”|-|-N*. Since

this inequality must hold for all real N, a contradiction is obtained by placing N = y

” ...

Page 1027

scalar A belongs to the spectrum of ET. Then, for some non-zero a in EX), we

have ETa' = Air. Then Ta' = Aw--y, where y belongs to the subspace (I–E)S), and

...

**Hence**A belongs to the spectrum of ET. Conversely, suppose that a non-zeroscalar A belongs to the spectrum of ET. Then, for some non-zero a in EX), we

have ETa' = Air. Then Ta' = Aw--y, where y belongs to the subspace (I–E)S), and

...

Page 1227

Proof. By Lemma 8(a), Q(T) is closed. Suppose {r,} is a sequence of elements of

Q, converging to re 3 (T"), then {a, , Tor,)} = {a, , ir,J} CI'(T") converges to [æ, ir] = [

a, Tor], since T" is closed.

...

Proof. By Lemma 8(a), Q(T) is closed. Suppose {r,} is a sequence of elements of

Q, converging to re 3 (T"), then {a, , Tor,)} = {a, , ir,J} CI'(T") converges to [æ, ir] = [

a, Tor], since T" is closed.

**Hence**To r = ir, or a e 3), .**Hence**3), is closed. Similarly...

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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