Linear Operators: Spectral operators |
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Page 2256
... compact spectral sets of T is uniformly bounded , and ( b ) no non - zero x in X satisfies the equation E ( o ) x = 0 for every compact spectral set o of T. 1 PROOF . Suppose first that ( a ) and ( b ) are satisfied . Since σ ( T ) is ...
... compact spectral sets of T is uniformly bounded , and ( b ) no non - zero x in X satisfies the equation E ( o ) x = 0 for every compact spectral set o of T. 1 PROOF . Suppose first that ( a ) and ( b ) are satisfied . Since σ ( T ) is ...
Page 2357
... compact if P ( T - I ) is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of the theorem , we may consequently pass from consideration of the operators T and T + P to consideration of the operators S and S + ( N + P ) ...
... compact if P ( T - I ) is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of the theorem , we may consequently pass from consideration of the operators T and T + P to consideration of the operators S and S + ( N + P ) ...
Page 2360
... compact . From this , ( iii ) , and Theorem VI.5.4 , it will follow that B ( μ ) Ru ; T + P ) is compact for μ in V , and i sufficiently large , so that the theorem will be proved . i = Let μ be in V. To show that | T'R ( μ ; T ) A ...
... compact . From this , ( iii ) , and Theorem VI.5.4 , it will follow that B ( μ ) Ru ; T + P ) is compact for μ in V , and i sufficiently large , so that the theorem will be proved . i = Let μ be in V. To show that | T'R ( μ ; T ) A ...
Contents
SPECTRAL OPERATORS | 1924 |
14 | 1983 |
Sufficient Conditions | 2134 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary asymptotic B₁ Banach space Boolean algebra Borel set boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complete Boolean algebra complex numbers complex plane continuous functions converges Corollary countably additive Definition denote differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem follows immediately formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality inverse L₁ Lebesgue Math multiplicity Nauk SSSR norm operators in Hilbert perturbation PROOF properties prove quasi-nilpotent resolution Russian satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose trace class type spectral operator unbounded uniformly bounded vector zero