Linear Operators, Part 2 |
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Page 2064
... norm of ƒ . The algebra A1 is not complete as a subalge- bra of A , but if we renorm it by placing ( 6 ) | a | 1 = | α | + | f | 1 , then the completeness of L1 assures the completeness of A1 under the norm ( 6 ) and the inequality ( 5 ) ...
... norm of ƒ . The algebra A1 is not complete as a subalge- bra of A , but if we renorm it by placing ( 6 ) | a | 1 = | α | + | f | 1 , then the completeness of L1 assures the completeness of A1 under the norm ( 6 ) and the inequality ( 5 ) ...
Page 2070
... norm ( 16 ) | ƒ | 0 = \ ƒ | 1 + \\ ƒ || , fe Lo , where f is the norm in X. We let 2o consist of all operators in 5 which have the form ( 17 ) a = αe + f , fe Lo . Theorem 1 shows that the operator a determines a and ƒ uniquely , and so ...
... norm ( 16 ) | ƒ | 0 = \ ƒ | 1 + \\ ƒ || , fe Lo , where f is the norm in X. We let 2o consist of all operators in 5 which have the form ( 17 ) a = αe + f , fe Lo . Theorem 1 shows that the operator a determines a and ƒ uniquely , and so ...
Page 2462
... norm , and CT converges to zero in trace norm . n - .... PROOF . The set K = C ( { x = H || x | ≤ 1 } ) is conditionally compact , and thus for each ɛ > 0 there exists a finite set x1 , of elements of K Xm such that each x = K ...
... norm , and CT converges to zero in trace norm . n - .... PROOF . The set K = C ( { x = H || x | ≤ 1 } ) is conditionally compact , and thus for each ɛ > 0 there exists a finite set x1 , of elements of K Xm such that each x = K ...
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