Linear Operators: Spectral operators |
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Page 2225
... proved that a bounded monotone generalized sequence of projections in a reflexive space is strongly convergent to a projection . In the case of Hilbert spaces , Lemma 3.14 was proved by Wolf [ 2 ] . It was observed by F. Riesz [ 21 ] ...
... proved that a bounded monotone generalized sequence of projections in a reflexive space is strongly convergent to a projection . In the case of Hilbert spaces , Lemma 3.14 was proved by Wolf [ 2 ] . It was observed by F. Riesz [ 21 ] ...
Page 2459
... prove this , we reason as follows . We may write Hx Y1 + y2 , where y1 € Σac ( H ) , Y2 € Hac ( H ) . We have ( I + H2 ) − 1 — ( ¿ I — H ) − 1 ( ¿ I — H ) 1 , Theorem XII.2.6 , and therefore , using what we have already proved , we ...
... prove this , we reason as follows . We may write Hx Y1 + y2 , where y1 € Σac ( H ) , Y2 € Hac ( H ) . We have ( I + H2 ) − 1 — ( ¿ I — H ) − 1 ( ¿ I — H ) 1 , Theorem XII.2.6 , and therefore , using what we have already proved , we ...
Page 2462
... proving | T , C | → 0 as n → ∞ . n n By Theorem VI.5.2 , C * is a compact operator . Thus , by what we have already proved , | T , C * | → 0 as n → ∞ . But | CT * | = | ( T „ C * ) * | = | T „ C * | , so | CT → 0 as n → ∞ also ...
... proving | T , C | → 0 as n → ∞ . n n By Theorem VI.5.2 , C * is a compact operator . Thus , by what we have already proved , | T , C * | → 0 as n → ∞ . But | CT * | = | ( T „ C * ) * | = | T „ C * | , so | CT → 0 as n → ∞ also ...
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