## Linear Operators: Spectral theory |

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Page 1207

for which the spaces ,£)„, a e A, are

suffices to observe that no x ^ 0 is

x # 0 is

point y ...

for which the spaces ,£)„, a e A, are

**orthogonal**. Thus to prove the lemma itsuffices to observe that no x ^ 0 is

**orthogonal**to each of the spaces £)„. Indeed, ifx # 0 is

**orthogonal**to the space £)0 then, for a bounded Borel function F and apoint y ...

Page 1227

10 Lemma, (a) ^(T), and 3)_ are closed

space 55 (T*). (b) 25(3"*) = <$)(T) © $+ © S5_. Proof. By Lemma 8(a), ^)(T) is

closed. Suppose {xn} is a sequence of elements of 55+ converging to x e ^(T*),

then {[#„ ...

10 Lemma, (a) ^(T), and 3)_ are closed

**orthogonal**sub- spaces of the Hilbertspace 55 (T*). (b) 25(3"*) = <$)(T) © $+ © S5_. Proof. By Lemma 8(a), ^)(T) is

closed. Suppose {xn} is a sequence of elements of 55+ converging to x e ^(T*),

then {[#„ ...

Page 1262

Then there exists a Hilbert space ^ 2 and an

Ax = PQx, x e$Q, P denoting the

sequence of bounded operators in Hilbert space Then there exists a Hilbert

space ...

Then there exists a Hilbert space ^ 2 and an

**orthogonal**projection Q in such thatAx = PQx, x e$Q, P denoting the

**orthogonal**projection of §j on 29 Let {Tn} be asequence of bounded operators in Hilbert space Then there exists a Hilbert

space ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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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