Linear Operators, Part 2 |
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Page 2154
... manifolds { x | ( T — \ I ) " x = 0 } increase with m , it follows that -- ( T _AI ) n + 1 ¥ + { x | ( T_ \ I ) n + 1 = 0 } - is dense in X. By induction , it is seen that the manifold ( T − λ ) n + kX + { x | ( T — \ I ) n + kx = 0 } ...
... manifolds { x | ( T — \ I ) " x = 0 } increase with m , it follows that -- ( T _AI ) n + 1 ¥ + { x | ( T_ \ I ) n + 1 = 0 } - is dense in X. By induction , it is seen that the manifold ( T − λ ) n + kX + { x | ( T — \ I ) n + kx = 0 } ...
Page 2214
... manifold which is invariant under every member of B. The theorem shows that A is in the uniformly closed algebra ( B ) generated by B. Thus W ( B ) ≤ A ( B ) . On the other hand , it is clear that A ( B ) ≤ W ( B ) . Q.E.D. The ...
... manifold which is invariant under every member of B. The theorem shows that A is in the uniformly closed algebra ( B ) generated by B. Thus W ( B ) ≤ A ( B ) . On the other hand , it is clear that A ( B ) ≤ W ( B ) . Q.E.D. The ...
Page 2217
... manifold which remains invariant under every element of B. PROOF . Let B1 be the strong closure of B. By Lemma 23 ... manifold that is left invariant by every member of B. Since it is evident that a closed linear manifold is invariant ...
... manifold which remains invariant under every element of B. PROOF . Let B1 be the strong closure of B. By Lemma 23 ... manifold that is left invariant by every member of B. Since it is evident that a closed linear manifold is invariant ...
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