Linear Operators, Part 2 |
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Page 1257
... isometric transformation ( which is not necessarily everywhere defined ) . Show the following : The operator T is ... isometric operator such that I - V is one - to - one and has a dense range , equation [ †† ] defines a symmetric ...
... isometric transformation ( which is not necessarily everywhere defined ) . Show the following : The operator T is ... isometric operator such that I - V is one - to - one and has a dense range , equation [ †† ] defines a symmetric ...
Page 1272
... isometric operator V is unitary if and only if D ( V ) = H = R ( V ) , i.e. , if d 0 = d_ . Also it is clear that a closed isometric operator has a unitary extension if and only if there is an isometric mapping of D , onto D , and that ...
... isometric operator V is unitary if and only if D ( V ) = H = R ( V ) , i.e. , if d 0 = d_ . Also it is clear that a closed isometric operator has a unitary extension if and only if there is an isometric mapping of D , onto D , and that ...
Page 1373
... isometric isomorphism Ŵ of E ( A ) L2 ( I ) onto L2 ( A , { p ;; } ) . It follows similarly that this same limit exists in the topology of L¿ ( 4 , { ^ ;; } ) for each fe L2 ( I ) , and defines an isometric isomorphism of E ( A ) L ( I ) ...
... isometric isomorphism Ŵ of E ( A ) L2 ( I ) onto L2 ( A , { p ;; } ) . It follows similarly that this same limit exists in the topology of L¿ ( 4 , { ^ ;; } ) for each fe L2 ( I ) , and defines an isometric isomorphism of E ( A ) L ( I ) ...
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