Linear Operators: Spectral operators |
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Page 1953
... range of V is closed . Let y be in the closure of the range of V. Then for some sequence { x } in E ( { 0 } ' ) X_we have Vx → y and , since the range of T is closed , there is an x in X with Hence - Tx = y . VE ( { 0 } ' ) x = TE ...
... range of V is closed . Let y be in the closure of the range of V. Then for some sequence { x } in E ( { 0 } ' ) X_we have Vx → y and , since the range of T is closed , there is an x in X with Hence - Tx = y . VE ( { 0 } ' ) x = TE ...
Page 1954
... range if and only if = 0 is either in the resolvent set of T or an isolated ( i ) the point spectral point of T , and ( ii ) the operator TE ( { 0 } ) has a closed range . PROOF . Let T have a closed range . We have already proved ( i ) ...
... range if and only if = 0 is either in the resolvent set of T or an isolated ( i ) the point spectral point of T , and ( ii ) the operator TE ( { 0 } ) has a closed range . PROOF . Let T have a closed range . We have already proved ( i ) ...
Page 2312
... range of T , and hence for all y in the closure of the range of T. On the other hand , if y is not in the closure of the range of T , then by the Hahn - Banach theorem ( II.3.13 ) there exists a y * in * such that y * ( y ) 0 , y * Tz ...
... range of T , and hence for all y in the closure of the range of T. On the other hand , if y is not in the closure of the range of T , then by the Hahn - Banach theorem ( II.3.13 ) there exists a y * in * such that y * ( y ) 0 , y * Tz ...
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