Linear Operators: Spectral theory |
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Page 1126
1 1 į 0 of the closed set C ; we shall denote this subspace of L20 , 1 ] by the symbol s ( C ) . Since each projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of ...
1 1 į 0 of the closed set C ; we shall denote this subspace of L20 , 1 ] by the symbol s ( C ) . Since each projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of ...
Page 1635
Throughout the rest of the present chapter , the symbol J will denote an index , i.e. , a k - tuple J [ 11 , ... , Ik ] of integers . We write IJI = k , min J = min siskli , max J = max , siskli .
Throughout the rest of the present chapter , the symbol J will denote an index , i.e. , a k - tuple J [ 11 , ... , Ik ] of integers . We write IJI = k , min J = min siskli , max J = max , siskli .
Page 1636
In general , unless the contrary is explicitly stated , J , , I , etc. , will denote indices for E ” , that is , indices whose range of variation is restricted by the condition min J 21 , max J S n .
In general , unless the contrary is explicitly stated , J , , I , etc. , will denote indices for E ” , that is , indices whose range of variation is restricted by the condition min J 21 , max J S n .
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Contents
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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