Linear Operators, Part 2 |
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Page 2035
... semi - group for which A is the infinite- simal generator . The semi - group T ( t ) and the function q have the further properties : 810 ) ( iv ) The derivative q ' ( 0 ) exists and p ' ( 0 ) = Ap ( 0 ) if and only if ( 0 ) € D ( Ag ) ...
... semi - group for which A is the infinite- simal generator . The semi - group T ( t ) and the function q have the further properties : 810 ) ( iv ) The derivative q ' ( 0 ) exists and p ' ( 0 ) = Ap ( 0 ) if and only if ( 0 ) € D ( Ag ) ...
Page 2037
... semi - group on [ 0 , ∞ ) of bounded linear operators in HP . Let B be the infinitesimal generator of the semi - group Î ( t ) , 0 ≤t < ∞o . It will now be shown that BÂ . For e D ( B ) , we have , by definition , = Î ( h ) - Î lim 4 ...
... semi - group on [ 0 , ∞ ) of bounded linear operators in HP . Let B be the infinitesimal generator of the semi - group Î ( t ) , 0 ≤t < ∞o . It will now be shown that BÂ . For e D ( B ) , we have , by definition , = Î ( h ) - Î lim 4 ...
Page 2087
... semi - group S ( t ) , t≥ 0 , of bounded linear operators in H and S ( t ) has a strongly analytic extension to a semi - group S ( ) defined for ( in the half plane R ( Ž ) > 0 . For every ( 0 ) in Ho there is one and only one ...
... semi - group S ( t ) , t≥ 0 , of bounded linear operators in H and S ( t ) has a strongly analytic extension to a semi - group S ( ) defined for ( in the half plane R ( Ž ) > 0 . For every ( 0 ) in Ho there is one and only one ...
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