Linear Operators, Part 2 |
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Page 2170
... orthogonal vectors x , y satisfy the relation | xy | 2 x2 + y2 , it will suffice to show that x is orthogonal to y if their spectra σ ( x ) and σ ( y ) , relative to the self adjoint operator T , are disjoint . In this case the function ...
... orthogonal vectors x , y satisfy the relation | xy | 2 x2 + y2 , it will suffice to show that x is orthogonal to y if their spectra σ ( x ) and σ ( y ) , relative to the self adjoint operator T , are disjoint . In this case the function ...
Page 2216
... orthogonal projection onto Ho then , since E ( e ) leaves Ho invariant , PE ( e ) P = E ( e ) P . If y is orthogonal to So , ( E ( e ) y , Ho ) = ( y , E ( e ) H 。) = 0 , so that E ( e ) y is orthogonal to Ho . Thus E ( e ) leaves the ...
... orthogonal projection onto Ho then , since E ( e ) leaves Ho invariant , PE ( e ) P = E ( e ) P . If y is orthogonal to So , ( E ( e ) y , Ho ) = ( y , E ( e ) H 。) = 0 , so that E ( e ) y is orthogonal to Ho . Thus E ( e ) leaves the ...
Page 2459
... orthogonal . Statement ( b ) of our lemma follows at once . ас = - If an € Σac ( H ) and lim , x = x , then , by what we have already proved , we may write x Y1 + y2 + Y3 , where y1 € Lac ( H ) and y2 , Yз are orthogonal to Σac ( H ) ...
... orthogonal . Statement ( b ) of our lemma follows at once . ас = - If an € Σac ( H ) and lim , x = x , then , by what we have already proved , we may write x Y1 + y2 + Y3 , where y1 € Lac ( H ) and y2 , Yз are orthogonal to Σac ( H ) ...
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