Linear Operators, Part 2 |
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Page 1951
... belong to the right ( left ) ideal J in B ( X ) . Then every projection E ( o ) with 0 σ belongs to 3. If 3 is closed , then S and N also belong to J. σ PROOF . Let 0 ō and let T , = TE ( 0 ) | E ( o ) X , the restriction of T to the ...
... belong to the right ( left ) ideal J in B ( X ) . Then every projection E ( o ) with 0 σ belongs to 3. If 3 is closed , then S and N also belong to J. σ PROOF . Let 0 ō and let T , = TE ( 0 ) | E ( o ) X , the restriction of T to the ...
Page 2264
... belongs to the point ( respectively residual or continuous ) spectrum of S if and only if it belongs to the point ( respectively residual or continuous ) spectrum of S2 . Since S2 is a spectral operator of scalar type , we have ( S2 — v ...
... belongs to the point ( respectively residual or continuous ) spectrum of S if and only if it belongs to the point ( respectively residual or continuous ) spectrum of S2 . Since S2 is a spectral operator of scalar type , we have ( S2 — v ...
Page 2462
... belongs to the trace class . Then , by what we have already proved , T , A converges to zero in norm , and thus , by Lemma XI.9.9 , TC = ( TA ) B converges to zero in trace norm . By Lemma XI.9.6 ( c ) and Definition XI.9.1 , C * belongs ...
... belongs to the trace class . Then , by what we have already proved , T , A converges to zero in norm , and thus , by Lemma XI.9.9 , TC = ( TA ) B converges to zero in trace norm . By Lemma XI.9.6 ( c ) and Definition XI.9.1 , C * belongs ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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 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