Linear Operators, Part 2 |
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Page 970
... similar integral representation for this extension . Let & be the family of all Borel subsets of R with finite measure and direct by inclusion . If % , denotes the characteristic function of the set e in & , and if ƒ is in L¿ ( R ) ...
... similar integral representation for this extension . Let & be the family of all Borel subsets of R with finite measure and direct by inclusion . If % , denotes the characteristic function of the set e in & , and if ƒ is in L¿ ( R ) ...
Page 1250
... similar to that of Stieltjes and differs from it by using the whole real axis ( ∞ , ∞ ) instead of [ 0 , ∞ ) . The Hausdorff moment problem is again similar but refers to a finite interval of 1250 XII.8 XII . UNBOUNDED OPERATORS IN ...
... similar to that of Stieltjes and differs from it by using the whole real axis ( ∞ , ∞ ) instead of [ 0 , ∞ ) . The Hausdorff moment problem is again similar but refers to a finite interval of 1250 XII.8 XII . UNBOUNDED OPERATORS IN ...
Page 1434
... similar results can be obtained for singularities at infinity . Suppose that we are dealing with an equation of the form n Σxx ( z ) ( n - * ) ( * ) ( z ) = 0 , k = 0 [ ** ] where a ( z ) = 1 and the coefficients are analytic in the ...
... similar results can be obtained for singularities at infinity . Suppose that we are dealing with an equation of the form n Σxx ( z ) ( n - * ) ( * ) ( z ) = 0 , k = 0 [ ** ] where a ( z ) = 1 and the coefficients are analytic in the ...
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