Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |
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Page 888
Here we have used the notations A i B and A v B for the intersection and union of two commuting projections A and B. We ... Also the ranges of the intersection and union of two commuting projection operators are given by the equations ...
Here we have used the notations A i B and A v B for the intersection and union of two commuting projections A and B. We ... Also the ranges of the intersection and union of two commuting projection operators are given by the equations ...
Page 1123
We say that E is a subdiagonalizing projection for T if T leaves the range of E invariant , i.e. , if ETE = TE . ... Any operator T in Hilbert space admits a maximal totally ordered set F of orthogonal subdiagonalizing projections ...
We say that E is a subdiagonalizing projection for T if T leaves the range of E invariant , i.e. , if ETE = TE . ... Any operator T in Hilbert space admits a maximal totally ordered set F of orthogonal subdiagonalizing projections ...
Page 1126
Since each projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of the projections Ej , it follows from ( 1 ) that the closure in $ ( xm ) of the vectors ( 4 ) is ...
Since each projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations of the projections Ej , it follows from ( 1 ) that the closure in $ ( xm ) of the vectors ( 4 ) is ...
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additive adjoint operator algebra analytic assume B-algebra 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 eigenvalues element equal equation Exercise exists extension fact finite dimensional follows formal formal differential operator formula function function f given Hence Hilbert space Hilbert-Schmidt ideal identity independent inequality integral interval isometric isomorphism Lemma limit linear matrix measure multiplicity neighborhood norm normal operator obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solution spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero