Linear Operators, Part 2 |
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Page 1976
... integral ( ii ) Soc , P ( X ) E ( dλ ; Â ( 8 ) ) is an e - essentially bounded E - measurable function of s . The integral ( iii ) [ _E ( σ ; Â ( s ) ) e ( ds ) , σ € B , is a bounded , countably additive spectral measure in Ho and ( iv ) ...
... integral ( ii ) Soc , P ( X ) E ( dλ ; Â ( 8 ) ) is an e - essentially bounded E - measurable function of s . The integral ( iii ) [ _E ( σ ; Â ( s ) ) e ( ds ) , σ € B , is a bounded , countably additive spectral measure in Ho and ( iv ) ...
Page 1990
... integral , that is , whether the integral is an ordinary Lebesgue integral defined for almost all s , a Cauchy type principal value integral defined as a certain limit of Lebesgue integrals , an integral of a vector valued function , or ...
... integral , that is , whether the integral is an ordinary Lebesgue integral defined for almost all s , a Cauchy type principal value integral defined as a certain limit of Lebesgue integrals , an integral of a vector valued function , or ...
Page 1991
... integral ( 21 ) X ( s ) = √ e - istλ ( dt ) , SЄ RN , RN which will be used presently . The translation qt , however , is not compact valued , and we proceed as follows . For an H valued function f , bounded and continuous on RN , and ...
... integral ( 21 ) X ( s ) = √ e - istλ ( dt ) , SЄ RN , RN which will be used presently . The translation qt , however , is not compact valued , and we proceed as follows . For an H valued function f , bounded and continuous on RN , and ...
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