Linear Operators: Spectral operators |
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Page 1936
The unique countably additive resolution of the identity defined on the Borel sets in the plane which is determined by a spectral operator T is called the resolution of the identity for T. 10 THEOREM . Let I Elt + En where E1 , ...
The unique countably additive resolution of the identity defined on the Borel sets in the plane which is determined by a spectral operator T is called the resolution of the identity for T. 10 THEOREM . Let I Elt + En where E1 , ...
Page 2094
It follows from some results of Bade [ 4 ] ( see also XVII.2.1 and XVII.2.12 ) that if X is a weakly complete B - space , then any prespectral operator is automatically spectral , and so has a unique resolution of the identity .
It follows from some results of Bade [ 4 ] ( see also XVII.2.1 and XVII.2.12 ) that if X is a weakly complete B - space , then any prespectral operator is automatically spectral , and so has a unique resolution of the identity .
Page 2242
The projection valued measure E is said to be the resolution of the identity for T. 13 LEMMA . An unbounded spectral operator of scalar type in the sense of Definition 12 is a spectral operator in the sense of Definition 1.
The projection valued measure E is said to be the resolution of the identity for T. 13 LEMMA . An unbounded spectral operator of scalar type in the sense of Definition 12 is a spectral operator in the sense of Definition 1.
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Contents
SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |
Introduction | 1927 |
Terminology and Preliminary Notions | 1931 |
Copyright | |
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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero