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 2295
... linearly independent . Hence E ( o ; T ) X is at least k - dimen- sional . This proves the remainder of the present lemma . Q.E.D. 4 DEFINITION . Let T be an unbounded discrete operator in the B - space X , with spectrum { A } . If E ...
... linearly independent . Hence E ( o ; T ) X is at least k - dimen- sional . This proves the remainder of the present lemma . Q.E.D. 4 DEFINITION . Let T be an unbounded discrete operator in the B - space X , with spectrum { A } . If E ...
Page 2327
... linearly independent solutions P1 , P2 of the equation 79 = λoy such that B¡ ( 91 ) T ф B1 ( 92 ) = 0 , i = 1 , ... , k . Then 1 may be represented uniquely as P1 = Σ = 1C1 , 0 , ( μ ( A ) ) and , similarly , 92 = -1 C2 , 0 , ( u ( A ) ...
... linearly independent solutions P1 , P2 of the equation 79 = λoy such that B¡ ( 91 ) T ф B1 ( 92 ) = 0 , i = 1 , ... , k . Then 1 may be represented uniquely as P1 = Σ = 1C1 , 0 , ( μ ( A ) ) and , similarly , 92 = -1 C2 , 0 , ( u ( A ) ...
Page 2394
... linearly independent , we must have a linear relation σ2 = aô1 + bơ1 . It is clear from Lemma 1 that such a linear combination can only have the indicated asymptotic form if a = 1 , b = c ( μ ( λ ) ) . Hence σ1⁄2 ( t , μ ( λ ) ) = 01 ...
... linearly independent , we must have a linear relation σ2 = aô1 + bơ1 . It is clear from Lemma 1 that such a linear combination can only have the indicated asymptotic form if a = 1 , b = c ( μ ( λ ) ) . Hence σ1⁄2 ( t , μ ( λ ) ) = 01 ...
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