Linear Operators: Spectral operators |
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Page 2239
... follows from Lemma 6 that T ( f ) is a closed , densely defined operator . Moreover , statement ( g ) follows from Corollary 7 . Statement ( d ) is obvious . Letting e € 2。 and x = E ( e ) X , we have = Tfxe ) x lim T ( ƒxe ) E ( en ) ...
... follows from Lemma 6 that T ( f ) is a closed , densely defined operator . Moreover , statement ( g ) follows from Corollary 7 . Statement ( d ) is obvious . Letting e € 2。 and x = E ( e ) X , we have = Tfxe ) x lim T ( ƒxe ) E ( en ) ...
Page 2246
... follows from Corollary 4 and Theorem 9 ( i ) that F com- mutes with E. By Theorem 9 ( ii ) , the restriction ( sin C ) ... follows that ( sin #C ) ” P ( n ) H = { 0 } . Let e be a closed set in the complex plane not containing zero . Then ...
... follows from Corollary 4 and Theorem 9 ( i ) that F com- mutes with E. By Theorem 9 ( ii ) , the restriction ( sin C ) ... follows that ( sin #C ) ” P ( n ) H = { 0 } . Let e be a closed set in the complex plane not containing zero . Then ...
Page 2459
... follows at once . If xn € Σac ( H ) and lim∞ = x , then , by what we have already proved , we may write x = y1 + y2 + ys , where y1 E Zac ( H ) and yY2 , Yз are orthogonal to Σac ( H ) . But , since x , € Σac ( H ) we have ( în , y ) ...
... follows at once . If xn € Σac ( H ) and lim∞ = x , then , by what we have already proved , we may write x = y1 + y2 + ys , where y1 E Zac ( H ) and yY2 , Yз are orthogonal to Σac ( H ) . But , since x , € Σac ( H ) we have ( în , y ) ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary B-algebra B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain eigenvalues elements equation exists finite number follows from Lemma follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial PROOF properties prove quasi-nilpotent resolution Russian S₁ satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset subspace Suppose trace class type spectral operator unbounded uniformly bounded unique vector zero