Linear Operators: Spectral operators |
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Page 2070
... norm ( 16 ) \ ƒ│0 = \ ƒ│1 + || ƒ || , fe Lo , where f is the norm in X. We let Ao consist of all operators in H which have the form ( 17 ) a = xe + f , fe Lo . α 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 Ao consist of all operators in H which have the form ( 17 ) a = xe + f , fe Lo . α Theorem 1 shows that the operator a determines a and ƒ uniquely , and so ...
Page 2450
... norm of an element A e 2 is simply the Hilbert - Schmidt norm of the operator RA . - - If A , € A and { A } is a Cauchy sequence , then , since R - 1 is bounded , | ( An — Am ) x | ≤ || A , — A „ || ≤ | R − 1 | || R ( A „ — A ...
... norm of an element A e 2 is simply the Hilbert - Schmidt norm of the operator RA . - - If A , € A and { A } is a Cauchy sequence , then , since R - 1 is bounded , | ( An — Am ) x | ≤ || A , — A „ || ≤ | R − 1 | || R ( A „ — A ...
Page 2462
... norm , and CT converges to zero in trace norm . - ... " PROOF . The set K = C ( { x = H || x | ≤ 1 } ) is conditionally compact , and thus for each ɛ > 0 there exists a finite set x1 , xm of elements of K such that each x = K satisfies ...
... norm , and CT converges to zero in trace norm . - ... " PROOF . The set K = C ( { x = H || x | ≤ 1 } ) is conditionally compact , and thus for each ɛ > 0 there exists a finite set x1 , xm of elements of K such that each x = K satisfies ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
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A₁ algebra Amer analytic applications arbitrary B-space Banach Banach space Boolean algebra Borel sets boundary bounded Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator Doklady Akad elements equation equivalent established example exists extension finite follows formula function given gives H₁ Hence Hilbert space hypothesis identity integral invariant inverse Lemma limit linear operators Math multiplicity Nauk SSSR norm normal perturbation plane positive preceding present problem Proc projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero