Linear Operators: Spectral operators |
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Page 2245
... orthonormal basis ( x ( ) , ... , xn ) . Let A ( n ) xnx for xen , and let B ( n ) be defined by B ( n ) x ( n ) 2 - 1nx { " 1 , 1 < i≤n , B ( n ) ( n ) = 0. Let 5 be the direct sum = 1 ( n ) · Let A and B be the operators in H defined ...
... orthonormal basis ( x ( ) , ... , xn ) . Let A ( n ) xnx for xen , and let B ( n ) be defined by B ( n ) x ( n ) 2 - 1nx { " 1 , 1 < i≤n , B ( n ) ( n ) = 0. Let 5 be the direct sum = 1 ( n ) · Let A and B be the operators in H defined ...
Page 2425
... orthonormal basis for H. Then ( 38 ) ( 9 ( 3 ) , x ) = ( lim ( + ple 81 +8 = lim = 0 + 3 8∞ + S S + f ( o ) - σ ) do , xa D - 8 ( f ( o ) , xa ) do 0-8 = P S + ∞ + ∞ 8 - σ ( f ( o ) , xa ) ( f ( o ) , xa ) do + ∞ = P 81 8 - σ do - [ ] ...
... orthonormal basis for H. Then ( 38 ) ( 9 ( 3 ) , x ) = ( lim ( + ple 81 +8 = lim = 0 + 3 8∞ + S S + f ( o ) - σ ) do , xa D - 8 ( f ( o ) , xa ) do 0-8 = P S + ∞ + ∞ 8 - σ ( f ( o ) , xa ) ( f ( o ) , xa ) do + ∞ = P 81 8 - σ do - [ ] ...
Page 2451
... orthonormal basis , and by ( 11 ) we have ( 16 ) ( TT ( A ) — F ( A ) T ) x = Σ ( Ax , xn ) xn — ŋ ( A ) x n = 1 - = Ax ― n ( A ) x . This shows that hypotheses ( a ) , ( b XX.3.1 2451 FRIEDRICHS ' METHOD FOR THE DISCRETE SPECTRUM.
... orthonormal basis , and by ( 11 ) we have ( 16 ) ( TT ( A ) — F ( A ) T ) x = Σ ( Ax , xn ) xn — ŋ ( A ) x n = 1 - = Ax ― n ( A ) x . This shows that hypotheses ( a ) , ( b XX.3.1 2451 FRIEDRICHS ' METHOD FOR THE DISCRETE SPECTRUM.
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
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 follows from Theorem 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