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 2174
... Hence we have 2 | ett ( S + T ) | = | ettseitт | ≤ | eits || ettT | ≤ M1M1⁄2 for all te R. Hence if this boundedness implied that S + T was a spectral operator , we would have a contradiction to McCarthy's [ 2 , I ] modification of ...
... Hence we have 2 | ett ( S + T ) | = | ettseitт | ≤ | eits || ettT | ≤ M1M1⁄2 for all te R. Hence if this boundedness implied that S + T was a spectral operator , we would have a contradiction to McCarthy's [ 2 , I ] modification of ...
Page 2308
... Hence we have only to show that T1 ( 7 ) - I maps D ( T1 ( 7 ) ) onto L2 ( I ) . Since A σ ( 7 ) , it follows that ( T1 ( 7 ) — λ ) D ( 7 ) is closed . Hence , in view of Lemma XII.1.6 ( d ) , it suffices to show that { ( T1 ( 7 ) — \ I ) ...
... Hence we have only to show that T1 ( 7 ) - I maps D ( T1 ( 7 ) ) onto L2 ( I ) . Since A σ ( 7 ) , it follows that ( T1 ( 7 ) — λ ) D ( 7 ) is closed . Hence , in view of Lemma XII.1.6 ( d ) , it suffices to show that { ( T1 ( 7 ) — \ I ) ...
Page 2357
... Hence - ( P + N ) ( S — \ I ) - ' = P ( S — XI ) - ' + N ( S — XI ) − ' = P ( T – XI ) - L + N ( S –AI ) - V is a bounded operator which is compact if P ( T — XI ) - ' is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of ...
... Hence - ( P + N ) ( S — \ I ) - ' = P ( S — XI ) - ' + N ( S — XI ) − ' = P ( T – XI ) - L + N ( S –AI ) - V is a bounded operator which is compact if P ( T — XI ) - ' is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of ...
Contents
SPECTRAL OPERATORS | 1924 |
An Operational Calculus for Bounded Spectral | 1941 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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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 inverse Krein L₁ Lemma locally convex spaces multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proc PROOF properties prove Pure Appl quasi-nilpotent resolution Russian S₁ 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