Linear Operators: Spectral operators |
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Page 1939
... quasi - nilpotent if and only if o ( T ) = { 0 } . PROOF . This follows from Lemma VII.3.4 . Q.E.D. 4 LEMMA . If S and N are bounded commuting operators and if N is quasi - nilpotent , then o ( S + N ) = σ ( S ) . PROOF . This is a ...
... quasi - nilpotent if and only if o ( T ) = { 0 } . PROOF . This follows from Lemma VII.3.4 . Q.E.D. 4 LEMMA . If S and N are bounded commuting operators and if N is quasi - nilpotent , then o ( S + N ) = σ ( S ) . PROOF . This is a ...
Page 2092
... quasi - nilpotent equivalent ; the general notion extends this case . ) The relation of being quasi - nilpotent equivalent is indeed an equivalence relation and , when T and U are quasi - nilpotent equivalent , then ( i ) o ( T ) = o ...
... quasi - nilpotent equivalent ; the general notion extends this case . ) The relation of being quasi - nilpotent equivalent is indeed an equivalence relation and , when T and U are quasi - nilpotent equivalent , then ( i ) o ( T ) = o ...
Page 2252
... quasi - nilpotent restriction to each space E ( o ) X with o bounded , need not be quasi - nilpotent itself . It may be bounded and not quasi - nilpotent , and it may even be unbounded . It is even possible that NS - 1 should fail to be ...
... quasi - nilpotent restriction to each space E ( o ) X with o bounded , need not be quasi - nilpotent itself . It may be bounded and not quasi - nilpotent , and it may even be unbounded . It is even possible that NS - 1 should fail to be ...
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