Linear Operators: Spectral theory |
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Page 1338
... respect to a positive o - finite measure μ . If the matrix of densities { m } is defined by the equations μ1 ( e ) ... respect to which all the set functions μ ,, are absolutely continuous . If { m } denotes the matrix of densities of { u ...
... respect to a positive o - finite measure μ . If the matrix of densities { m } is defined by the equations μ1 ( e ) ... respect to which all the set functions μ ,, are absolutely continuous . If { m } denotes the matrix of densities of { u ...
Page 1340
... respect to which the set functions Mij are continuous . Let { m } be the corresponding matrix of densities , and let { n , } be the matrix of densities of the Mij with respect to the measure ( μ + μ ) . If m is the density of u with respect ...
... respect to which the set functions Mij are continuous . Let { m } be the corresponding matrix of densities , and let { n , } be the matrix of densities of the Mij with respect to the measure ( μ + μ ) . If m is the density of u with respect ...
Page 1738
... respect to x , that + √y_h ( x ) { g ( x ) dx = ( −1 ) 3 ƒ „ îf h ( x ) g ( x ) dx + + for all g in Co ° ( V ) and h in Co ( √ ) , and it follows by continuity , since Co ( V ) is by definition of Hg ( V ) dense in Hö ( V ) , that ...
... respect to x , that + √y_h ( x ) { g ( x ) dx = ( −1 ) 3 ƒ „ îf h ( x ) g ( x ) dx + + for all g in Co ° ( V ) and h in Co ( √ ) , and it follows by continuity , since Co ( V ) is by definition of Hg ( V ) dense in Hö ( V ) , that ...
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