Linear Operators, Part 2 |
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Page 2291
... resolvent occurs so frequently in this section , it will be convenient to introduce , in the following definition , a special term for such operators . 1 DEFINITION . An operator T is discrete if there is a number λ in its resolvent set ...
... resolvent occurs so frequently in this section , it will be convenient to introduce , in the following definition , a special term for such operators . 1 DEFINITION . An operator T is discrete if there is a number λ in its resolvent set ...
Page 2316
... resolvent . By Lemma 8 , the condition for this is ( ❤n , n ) = 0 , where a solution of is ( T * − Xn I * ) & n = 0 . - Since by Theorems XIII.2.10 and XII.4.28 , T * is defined by the formal differential operator ( d / dt ) 2 and the ...
... resolvent . By Lemma 8 , the condition for this is ( ❤n , n ) = 0 , where a solution of is ( T * − Xn I * ) & n = 0 . - Since by Theorems XIII.2.10 and XII.4.28 , T * is defined by the formal differential operator ( d / dt ) 2 and the ...
Page 2363
... resolvent R ( λ ; T + P ) corresponding to one - dimensional eigenspaces if any one of the following conditions holds : ( a ) μ , approaches zero ; ( b ) lim supi → μ1 ≤ K < ∞ , and | P ( T — λ 。 I ) - ' | < 8 = 8 ( K , T ) ; - ( c ) ...
... resolvent R ( λ ; T + P ) corresponding to one - dimensional eigenspaces if any one of the following conditions holds : ( a ) μ , approaches zero ; ( b ) lim supi → μ1 ≤ K < ∞ , and | P ( T — λ 。 I ) - ' | < 8 = 8 ( K , T ) ; - ( c ) ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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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