Linear Operators, Part 2 |
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Page 2183
... clearly a subalgebra of A ( 7 ) . Let B denote its closure in the uniform topology of operators . By Theorem XV.4.5 and the fact that a scalar type operator is clearly in the uniformly closed algebra generated by the projections in its ...
... clearly a subalgebra of A ( 7 ) . Let B denote its closure in the uniform topology of operators . By Theorem XV.4.5 and the fact that a scalar type operator is clearly in the uniformly closed algebra generated by the projections in its ...
Page 2271
... clearly G and y * satisfy ( ii ) and ( iii ) . Let H = | \ { E \ y * Ex = y * x , x € M ( x 。) } . Clearly G≥ H. However , if y * Ex = y * x for all xe M ( x ) , then z * GEx = y * Ex = y * x = F * G * z * x = G * z * Fx = 2 * Fx , x ...
... clearly G and y * satisfy ( ii ) and ( iii ) . Let H = | \ { E \ y * Ex = y * x , x € M ( x 。) } . Clearly G≥ H. However , if y * Ex = y * x for all xe M ( x ) , then z * GEx = y * Ex = y * x = F * G * z * x = G * z * Fx = 2 * Fx , x ...
Page 2332
... clearly sufficient for us to show that for each ƒ in L2 ( 0 , ∞ ) , the sequence bm ( f ) defined for m ≥1 by the formula , 00 bm ( ƒ ) = √ e - mt f ( t ) dt 0 is in l2 . We may clearly suppose without loss of generality that ƒ is ...
... clearly sufficient for us to show that for each ƒ in L2 ( 0 , ∞ ) , the sequence bm ( f ) defined for m ≥1 by the formula , 00 bm ( ƒ ) = √ e - mt f ( t ) dt 0 is in l2 . We may clearly suppose without loss of generality that ƒ is ...
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