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 2357
... compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of the theorem , we may consequently pass from ... compact , it follows from Lemma VI.5.3 that for v > 0 the operator ( T — λ 。 I ) is compact . Thus , if v > 0 , then since P ...
... compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of the theorem , we may consequently pass from ... compact , it follows from Lemma VI.5.3 that for v > 0 the operator ( T — λ 。 I ) is compact . Thus , if v > 0 , then since P ...
Page 2360
... compact . From this , ( iii ) , and Theorem VI.5.4 , it will follow that B ( μ ) : R ( μ ; T + P ) is compact for μ in V , and i sufficiently large , so that the theorem will be proved . = Let μ be in V. To show that TR ( μ ; T ) A ...
... compact . From this , ( iii ) , and Theorem VI.5.4 , it will follow that B ( μ ) : R ( μ ; T + P ) is compact for μ in V , and i sufficiently large , so that the theorem will be proved . = Let μ be in V. To show that TR ( μ ; T ) A ...
Page 2462
... compact . Put CQR1 , and D QR1 , and D = R2 , so that VCD . The operator C is compact by Corollary VI.5.5 , and thus proof of Corollary 11 is complete . Q.E.D. 12 LEMMA . If C is a compact operator in H , and { T } is a uniformly ...
... compact . Put CQR1 , and D QR1 , and D = R2 , so that VCD . The operator C is compact by Corollary VI.5.5 , and thus proof of Corollary 11 is complete . Q.E.D. 12 LEMMA . If C is a compact operator in H , and { T } is a uniformly ...
Contents
SPECTRAL OPERATORS | 1924 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
The Algebras and | 1967 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary B-algebra 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 denote dense differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial PROOF properties prove quasi-nilpotent resolution Russian S₁ satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset subspace Suppose trace class type spectral operator unbounded uniformly bounded unique vector zero