Linear Operators: Spectral theory |
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Page 970
... limit in the norm of L ( R ) of the generalized sequence { f } . Hence , by Theorem 9 , 7f is the limit in the norm of L2 ( Mo ) of the generalized sequence { T ( Xef ) } . Equivalently , we write τή = lim [ [ x , f ( x ) dx , e where the ...
... limit in the norm of L ( R ) of the generalized sequence { f } . Hence , by Theorem 9 , 7f is the limit in the norm of L2 ( Mo ) of the generalized sequence { T ( Xef ) } . Equivalently , we write τή = lim [ [ x , f ( x ) dx , e where the ...
Page 1124
... limit q ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and E , ≤ E. If E is the strong limit of E , then EE and ( E ) = q ( E ) . Thus , it follows as above that E = E. This ...
... limit q ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and E , ≤ E. If E is the strong limit of E , then EE and ( E ) = q ( E ) . Thus , it follows as above that E = E. This ...
Page 1129
... limit of the operators Ø , • m On the other hand , it follows from Theorem X.2.1 that belongs to the algebra A , m m so that is a limit of linear combinations of products of the operators E. Since the projections E , form a totally ...
... limit of the operators Ø , • m On the other hand , it follows from Theorem X.2.1 that belongs to the algebra A , m m so that is a limit of linear combinations of products of the operators E. Since the projections E , form a totally ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero