Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |
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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 2387
... belongs to the point spectrum of T , and is a pole of the resolvent of T. PROOF . Statements ( i ) and ( ii ) follow from Lemma 1 and formulas ( 2a ) and ( 2b ) . Let λo R { 0 } . If A ( ) = 0 , then σ ( , μ ( o ) ) is an eigenvector of ...
... belongs to the point spectrum of T , and is a pole of the resolvent of T. PROOF . Statements ( i ) and ( ii ) follow from Lemma 1 and formulas ( 2a ) and ( 2b ) . Let λo R { 0 } . If A ( ) = 0 , then σ ( , μ ( o ) ) is an eigenvector of ...
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 = ( T2A ) 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 = ( T2A ) 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 Spectrum of a Spectral Operator | 1955 |
The Algebras A and | 1966 |
Copyright | |
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A₁ Acad adjoint operator algebra of projections Amer analytic arbitrary B-algebra B-space B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition dense differential operator Dokl Doklady Akad eigenvalues elements equation exists formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis ibid identity inequality invariant subspaces inverse Krein L₁ Lemma locally convex spaces multiplicity Nauk SSSR nonselfadjoint norm normal operators operators in Hilbert perturbation Proc PROOF properties prove Pures Appl quasi-nilpotent resolution Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose topology trace class type spectral operator unbounded uniformly bounded vector zero