## Linear Operators: Spectral theory |

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Page 1190

An operator T may be symmetric without having a dense domain but if S)(T) is

dense so that T* is defined then the notion of symmetry is equivalent to the

inclusion T* 2 T. Of course if T is a bounded

the ...

An operator T may be symmetric without having a dense domain but if S)(T) is

dense so that T* is defined then the notion of symmetry is equivalent to the

inclusion T* 2 T. Of course if T is a bounded

**everywhere**defined operator thenthe ...

Page 1212

... 0 for X$e„. Then \ (Bn+1f)(X)F(X)M(dX) = f /(*)(4,+1 *□)(*)*(*). = JSB/(«)(F(r)g)(*

Md»). = /5b/(»)(^. *□)(»>(*) = f (BB/)^)F(A)^(*tt). Thus, (B„+1/)(A) = (fi„/)(A)/<-

almost

...

... 0 for X$e„. Then \ (Bn+1f)(X)F(X)M(dX) = f /(*)(4,+1 *□)(*)*(*). = JSB/(«)(F(r)g)(*

Md»). = /5b/(»)(^. *□)(»>(*) = f (BB/)^)F(A)^(*tt). Thus, (B„+1/)(A) = (fi„/)(A)/<-

almost

**everywhere**on en. Consequently (BJ)(X) = jsJ(s)W„+1(s,X)y(ds), feL^S^v)...

Page 1233

It follows from the boundedness of (T1— A/)-1 and Lemma 1.2 that ^((^-A/)-1) is

closed. We wish to show that 25 ((7\— XI)-1) is all of Hilbert space. Suppose that

this is true for some given X0. Then {T1— X,/)"1 = R(Xq) is an

...

It follows from the boundedness of (T1— A/)-1 and Lemma 1.2 that ^((^-A/)-1) is

closed. We wish to show that 25 ((7\— XI)-1) is all of Hilbert space. Suppose that

this is true for some given X0. Then {T1— X,/)"1 = R(Xq) is an

**everywhere**defined...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero