## Linear Operators: Spectral operators |

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Page 1912

exception , used from

concerning approximation of integrable functions by smooth functions . The

exception is Sobolev ' s theorem , which is used only in the proof of a corollary

unessential ...

exception , used from

**Chapter**XIV are elementary self - contained lemmasconcerning approximation of integrable functions by smooth functions . The

exception is Sobolev ' s theorem , which is used only in the proof of a corollary

unessential ...

Page 1915

19 was presented as a corollary to the preceding theorem , but since it is

essential for a number of the deep results in

applications to some of the most difficult problems of contemporary atomic

physics , we ...

19 was presented as a corollary to the preceding theorem , but since it is

essential for a number of the deep results in

**Chapter**XX which have importantapplications to some of the most difficult problems of contemporary atomic

physics , we ...

Page 1916

properties of the set of eigenvectors of a perturbed operator . These theorems

make assertions less specific than those obtained in the first sections of

XIX ...

**Chapter**XIX ends with a section which gives results concerning completenessproperties of the set of eigenvectors of a perturbed operator . These theorems

make assertions less specific than those obtained in the first sections of

**Chapter**XIX ...

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algebra of projections analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear closure commuting compact complete consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero