Linear Operators: Spectral theory |
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Page 970
... limit in the norm of L2 ( R ) of the generalized sequence { x.f } . Hence , by Theorem 9 , tf is the limit in the norm of L2 ( M ) of the generalized sequence { T ( Xef ) } . Equivalently , we write τί lim [ x , • ] f ( x ) dx , e e ƒЄ ...
... limit in the norm of L2 ( R ) of the generalized sequence { x.f } . Hence , by Theorem 9 , tf is the limit in the norm of L2 ( M ) of the generalized sequence { T ( Xef ) } . Equivalently , we write τί lim [ x , • ] f ( x ) dx , e e ƒЄ ...
Page 1124
... limit ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and EE . If E is the strong limit of En then EE and ( E∞ ) = q ( E ) . Thus , it follows as above that E∞ = E. This ...
... limit ( E ) , then it follows from what we have already proved that E , is an increasing sequence of projections and EE . If E is the strong limit of En then EE and ( E∞ ) = q ( E ) . Thus , it follows as above that E∞ = E. This ...
Page 1697
... limit in the norm of HP ) ( UI ) of a sequence { g } of functions in Co ( U1 ) , from which the present lemma ... limit in the norm of HP ( C ) of a sequence of elements of C ( C ) . Thus , by Lemma 3.23 , fo1 is the limit in the π ...
... limit in the norm of HP ) ( UI ) of a sequence { g } of functions in Co ( U1 ) , from which the present lemma ... limit in the norm of HP ( C ) of a sequence of elements of C ( C ) . Thus , by Lemma 3.23 , fo1 is the limit in the π ...
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BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero