Linear Operators: Spectral theory |
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Page 970
... limit in the norm of L2 ( R ) of the generalized sequence { zef } . Hence , by Theorem 9 , Tf is the limit in the norm of L2 ( Mo ) of the generalized sequence { T ( Xef ) } . Equivalently , we write f τή = lim • S [ x , f ( x ) dx , e ...
... limit in the norm of L2 ( R ) of the generalized sequence { zef } . Hence , by Theorem 9 , Tf is the limit in the norm of L2 ( Mo ) of the generalized sequence { T ( Xef ) } . Equivalently , we write f τή = lim • S [ x , f ( x ) dx , e ...
Page 976
... limit g ( t ,. .... tn ) = ( 2л ) - " / 2 lim م + N N → ∞J - N ... + N f ( x1 , ... , xn ) -N · exp { −i ( t11 + ... + tnxn ) } dæ1 ... dxn exists in the norm of L2 ( R " ) , and defines a unitary mapping in this space whose inverse ...
... limit g ( t ,. .... tn ) = ( 2л ) - " / 2 lim م + N N → ∞J - N ... + N f ( x1 , ... , xn ) -N · exp { −i ( t11 + ... + tnxn ) } dæ1 ... dxn exists in the norm of L2 ( R " ) , and defines a unitary mapping in this space whose inverse ...
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 E , then EE and ( E ) = q ( E ) . Thus , it follows as above that EE . This proves ...
... 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 E , then EE and ( E ) = q ( E ) . Thus , it follows as above that EE . This proves ...
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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero