Linear Operators, Part 2 |
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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 ) σ ( T ) = σ ...
... 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 ) σ ( T ) = σ ...
Page 2115
... nilpotent . ) It is proved that if T is decomposable and T and U are quasi - nilpotent equivalent , then U is decomposable . Moreover , if T and U are decomposable , then XT ( F ) = X ( F ) for all closed sets F if and only if T and U ...
... nilpotent . ) It is proved that if T is decomposable and T and U are quasi - nilpotent equivalent , then U is decomposable . Moreover , if T and U are decomposable , then XT ( F ) = X ( F ) for all closed sets F if and only if T and U ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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 Borel function 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 eigenvalues elements equation exists finite number follows from Lemma 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