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 |
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