Linear Operators: Spectral operators |
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Page 2236
... statement ( v ) remains to be proved . In the first case described in statement ( v ) , σ ( T ) is a compact subset of U , and by Corollary XV.3.5 , E ( σ ( T ) ) = I so that without loss of generality each of the sets e , n ≥ 1 , may ...
... statement ( v ) remains to be proved . In the first case described in statement ( v ) , σ ( T ) is a compact subset of U , and by Corollary XV.3.5 , E ( σ ( T ) ) = I so that without loss of generality each of the sets e , n ≥ 1 , may ...
Page 2239
... Statement ( d ) is obvious . Letting e € 2。 and x = E ( e ) X , we have = Tfxe ) x lim T ( ƒxe ) E ( en ) x = lim T ... Statement ( a ) clearly will follow from statement ( b ) . Statement ( b ) follows from statement ( f ) ; indeed ...
... Statement ( d ) is obvious . Letting e € 2。 and x = E ( e ) X , we have = Tfxe ) x lim T ( ƒxe ) E ( en ) x = lim T ... Statement ( a ) clearly will follow from statement ( b ) . Statement ( b ) follows from statement ( f ) ; indeed ...
Page 2476
... statement ( 76 ) holds for each ge H ′ such that g1 ( a ) = 0 for all a ¤ ¤ ̧ . - Since ( H + V , H ) is a closed subspace of H ' ( cf. Lemma 2 ) it follows that every ğ Є H ' such that g1 ( a ) = 0 for all a € e , belongs to Σ ( H + V ...
... statement ( 76 ) holds for each ge H ′ such that g1 ( a ) = 0 for all a ¤ ¤ ̧ . - Since ( H + V , H ) is a closed subspace of H ' ( cf. Lemma 2 ) it follows that every ğ Є H ' such that g1 ( a ) = 0 for all a € e , belongs to Σ ( H + V ...
Contents
SPECTRAL OPERATORS | 1924 |
Introduction | 1927 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer arbitrary B*-algebra B₁ Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator Colojoară commuting compact complex numbers complex plane contains converges Corollary countably additive Definition dense differential operator disjoint Doklady Akad E-measurable eigenvalues elements equation equivalent exists Foias follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math matrix multiplicity norm operators in Hilbert perturbation polynomial PROOF proved quasi-nilpotent resolution restriction Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset subspace sufficiently type spectral operator unbounded unique vector weakly complete zero