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 1930
... countably additive . However , in Corollary X.2.4 we have seen that a bounded normal operator in Hilbert space always has a uniquely defined bounded and countably additive resolution ... countably additive 1930 XV.2.2 XV . SPECTRAL OPERATORS.
... countably additive . However , in Corollary X.2.4 we have seen that a bounded normal operator in Hilbert space always has a uniquely defined bounded and countably additive resolution ... countably additive 1930 XV.2.2 XV . SPECTRAL OPERATORS.
Page 2143
... countably additive on S ( T ) , and com- mutes with T. : PROOF . From Definitions 1 , 4 , and 7 it is seen that ( T ) S1 ( T ) , and thus for each 8 in ( T ) there is , by Lemma 2 , one and only one projection E ( S ) with E ( 8 ) x = x ...
... countably additive on S ( T ) , and com- mutes with T. : PROOF . From Definitions 1 , 4 , and 7 it is seen that ( T ) S1 ( T ) , and thus for each 8 in ( T ) there is , by Lemma 2 , one and only one projection E ( S ) with E ( 8 ) x = x ...
Page 2144
... countably additive in the X * , and is bounded . It remains only to show that A ( o ) A ( S ) A ( od ) . = topology of It is seen , by using the above remarks , that for a fixed σ the family of d for which the equation is valid is a o ...
... countably additive in the X * , and is bounded . It remains only to show that A ( o ) A ( S ) A ( od ) . = topology of It is seen , by using the above remarks , that for a fixed σ the family of d for which the equation is valid is a o ...
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