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 2152
... prove that M ( da ) is closed . In other words , we may and shall assume that 8 is the complement of an open subinterval y of To . Let { x } be a sequence in X , convergent to the point x , and with p ( x ) = y . To prove ( C ) it will ...
... prove that M ( da ) is closed . In other words , we may and shall assume that 8 is the complement of an open subinterval y of To . Let { x } be a sequence in X , convergent to the point x , and with p ( x ) = y . To prove ( C ) it will ...
Page 2190
... prove the final inequality of the present theorem it will suffice to prove that f│E≤S ( ƒ ) . Since both terms in this inequality are continuous functions of f , it will suffice to prove it for every function f in a set dense in EB ...
... prove the final inequality of the present theorem it will suffice to prove that f│E≤S ( ƒ ) . Since both terms in this inequality are continuous functions of f , it will suffice to prove it for every function f in a set dense in EB ...
Page 2236
... proves the first part of ( v ) . To prove the second part of ( v ) it is sufficient , since f ( T ) is closed , to show that f ( T ) is everywhere defined ( cf. the closed graph theorem ( II.2.4 ) ) . By ( ii ) , D ( ƒ ( T ) ) ≥ E ( e ) ...
... proves the first part of ( v ) . To prove the second part of ( v ) it is sufficient , since f ( T ) is closed , to show that f ( T ) is everywhere defined ( cf. the closed graph theorem ( II.2.4 ) ) . By ( ii ) , D ( ƒ ( T ) ) ≥ E ( e ) ...
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