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 has a unitary extension if and only if there is an isometric mapping of D , onto D , and that this can happen if and only if d + = d_ . + This elegant extension procedure is due to von Neumann [ 7 ] , and has also ...
... isometric operator has a unitary extension if and only if there is an isometric mapping of D , onto D , and that this can happen if and only if d + = d_ . + This elegant extension procedure is due to von Neumann [ 7 ] , and has also ...
Page 1373
... isometric isomorphisms onto all of L2 ( 4 , { ^ ,, } ) and L2 ( 4 , { μ , } ) , respectively . Since A f ( t ) ô , 4 ... isometric isomorphism Ŵ of E ( A ) L2 ( I ) onto L¿ ( A , { ^ ;; } ) . It follows similarly that this same limit ...
... isometric isomorphisms onto all of L2 ( 4 , { ^ ,, } ) and L2 ( 4 , { μ , } ) , respectively . Since A f ( t ) ô , 4 ... isometric isomorphism Ŵ of E ( A ) L2 ( I ) onto L¿ ( A , { ^ ;; } ) . It follows similarly that this same limit ...
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 domain 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 linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma 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