Linear Operators: Spectral operators |
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Page 2064
... norm of f . The algebra 2 , is not complete as a subalge- bra of A , but if we renorm it by placing ( 6 ) | a | 1 = | a | + | ƒ│1 , then the completeness of L1 assures the completeness of A1 under the norm ( 6 ) and the inequality ( 5 ) ...
... norm of f . The algebra 2 , is not complete as a subalge- bra of A , but if we renorm it by placing ( 6 ) | a | 1 = | a | + | ƒ│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 Дo consist of all operators in H which have the form ( 17 ) a = xe + f , fe Lo Theorem 1 shows that the operator a determines a and ƒ uniquely , and so we ...
... norm ( 16 ) | ƒ│0 = | ƒ│1 + || ƒ || , fe Lo , where f is the norm in X. We let Дo consist of all operators in H which have the form ( 17 ) a = xe + f , fe Lo Theorem 1 shows that the operator a determines a and ƒ uniquely , and so we ...
Page 2450
... norm of an element A e 2 is simply the Hilbert - Schmidt norm of the operator RA . n 1 n If An Є A and { A } is a Cauchy sequence , then , since R - 1 is bounded , | ( An — Am ) x | ≤ || An — Am || ≤ | R − 1 | || R ( Aŋ — Am ) ...
... norm of an element A e 2 is simply the Hilbert - Schmidt norm of the operator RA . n 1 n If An Є A and { A } is a Cauchy sequence , then , since R - 1 is bounded , | ( An — Am ) x | ≤ || An — Am || ≤ | R − 1 | || R ( Aŋ — Am ) ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
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A₁ adjoint operator 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 operators 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 sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero