## Linear Operators: Spectral theory |

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Page 861

Clearly if x*1

*> (T*-ly)* = T-Hyz), and if a = T~^le, then az = T~xz for every z e X. Also xa = Txa

= e = T?(Tme) = Tx\ex) = (T'^x = ax. Thus ar1

Clearly if x*1

**exists**then Tx-iTx = TXTX-X =1. If Txl**exists**in B(£), then T.\kT?m = y*> (T*-ly)* = T-Hyz), and if a = T~^le, then az = T~xz for every z e X. Also xa = Txa

= e = T?(Tme) = Tx\ex) = (T'^x = ax. Thus ar1

**exists**and = x~xz. 2 Definition.Page 1057

Thus (2) gives F(JST * /)(«) = (2*)-/* lim ^ f ^ ( f d» = (lim & f J£ jfc(i,)*""dy) *(/)(«).

provided only that the limit in the braces in this last equation

complete the proof of the present lemma, it suffices to show that (3) Q{u)=0>\ Sy^

et**dy ...

Thus (2) gives F(JST * /)(«) = (2*)-/* lim ^ f ^ ( f d» = (lim & f J£ jfc(i,)*""dy) *(/)(«).

provided only that the limit in the braces in this last equation

**exists**. Thus, tocomplete the proof of the present lemma, it suffices to show that (3) Q{u)=0>\ Sy^

et**dy ...

Page 1261

23 If an operator T has a closed linear extension there

linear extension T such that if Tx is any closed linear extension of T then T Q TY .

T is called the closure of T. (a) There

23 If an operator T has a closed linear extension there

**exists**a unique closedlinear extension T such that if Tx is any closed linear extension of T then T Q TY .

T is called the closure of T. (a) There

**exists**a densely defined operator with no ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero