Linear Operators: Spectral theory |
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Page 907
n8 a positive if and only if its spectrum lies on the unit circle , the real axis , or the non - negative real axis respectively . Proor . If N is a bounded normal operator then , by Corollary IX.3.15 , NN * = N * N = I if and only if a ...
n8 a positive if and only if its spectrum lies on the unit circle , the real axis , or the non - negative real axis respectively . Proor . If N is a bounded normal operator then , by Corollary IX.3.15 , NN * = N * N = I if and only if a ...
Page 1247
Q.E.D. , Next we shall require some information on positive self adjoint transformations and their square roots . = 2 LEMMA . A self adjoint transformation T is positive if and only if o ( T ) is a subset of the interval ( 0 , 0 ) .
Q.E.D. , Next we shall require some information on positive self adjoint transformations and their square roots . = 2 LEMMA . A self adjoint transformation T is positive if and only if o ( T ) is a subset of the interval ( 0 , 0 ) .
Page 1338
Let { uis } be a positive matrix measure whose elements Mis are continuous with respect to a positive o - finite measure u . If the matrix of densities { m } is defined by the equations Möjle ) = S.m. , ( 2 ) u ( da ) , where e is any ...
Let { uis } be a positive matrix measure whose elements Mis are continuous with respect to a positive o - finite measure u . If the matrix of densities { m } is defined by the equations Möjle ) = S.m. , ( 2 ) u ( da ) , where e is any ...
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Contents
8 | 876 |
859 | 885 |
extensive presentation of applications of the spectral theorem | 911 |
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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero