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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Results 1-3 of 91
Page 1983
... preceding corollary shows that A is a spectral operator . Since  ( s ) has distinct eigenvalues , it is a scalar operator , that is , its radical part is zero . Thus Corollary 9 shows that A is also a scalar type operator . Q.E.D. 11 ...
... preceding corollary shows that A is a spectral operator . Since  ( s ) has distinct eigenvalues , it is a scalar operator , that is , its radical part is zero . Thus Corollary 9 shows that A is also a scalar type operator . Q.E.D. 11 ...
Page 2232
... preceding theorem it follows that E ( e ) D ( Q ) = D ( Q ) and QE ( e ) x = E ( e ) Qx for every x in D ( Q ) and every e in Σ . PROOF . Let { e } be as in the preceding proof , and let x be in D ( Q ) . Then lim , E ( e ) E ( e „ ) x ...
... preceding theorem it follows that E ( e ) D ( Q ) = D ( Q ) and QE ( e ) x = E ( e ) Qx for every x in D ( Q ) and every e in Σ . PROOF . Let { e } be as in the preceding proof , and let x be in D ( Q ) . Then lim , E ( e ) E ( e „ ) x ...
Page 2455
... preceding lemma be H2 , H1 , H1 , we obtain the present corollary . Q.E.D. 5 COROLLARY . Under the hypotheses of the preceding corollary , U ( H1 , H2 ) is an isometric mapping of Σ ( H1 , H2 ) onto Σ ( H2 , H1 ) . PROOF . By the ...
... preceding lemma be H2 , H1 , H1 , we obtain the present corollary . Q.E.D. 5 COROLLARY . Under the hypotheses of the preceding corollary , U ( H1 , H2 ) is an isometric mapping of Σ ( H1 , H2 ) onto Σ ( H2 , H1 ) . PROOF . By the ...
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