## Linear Operators: Spectral theory |

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Page 917

[z.(W7), X.(Ws), . . .]. Now so = u(e) > 0, but vio – Š s z.(2)(

. s (

proves that Au o Pi. It will next be shown by induction that u(e,4é,) = 0 =

[z.(W7), X.(Ws), . . .]. Now so = u(e) > 0, but vio – Š s z.(2)(

**Vs**),(2)”; (d) n=1 */ en - S. s (

**vs**.(2)”; (d) = 0, n=1 Jen she *hich contradicts the fact that**V**is an isometry andproves that Au o Pi. It will next be shown by induction that u(e,4é,) = 0 =

**fi**(e ...Page 1255

...

follows from the Schwarz inequality that

define a mapping U(t): $8 -- $8 by letting U(t)r =

...

**V**(t)q) = X m(s1–s,)f(s1–t)g(se—t) 81, 82 - X m(s) +t–s, t)f(s)p(s) 81, 82 = (f, g), itfollows from the Schwarz inequality that

**V**(t)?so CŞso. Consequently, we candefine a mapping U(t): $8 -- $8 by letting U(t)r =

**V**(t)**f-i**-Qso, if r = f_HQso is in Q3.Page 1345

Similarly, my (A) = (M(A)w, us) -: a,(2)a(7)(M(2),(2),

of other elementary measure theoretic properties which these spaces have in

common with the spaces L2(u). However, a few words of caution are in order. If [

, ...

Similarly, my (A) = (M(A)w, us) -: a,(2)a(7)(M(2),(2),

**v**,(A)) –3,000 of A e N(20). ...of other elementary measure theoretic properties which these spaces have in

common with the spaces L2(u). However, a few words of caution are in order. If [

**fi**, ...

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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