Linear Operators: Spectral theory |
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Page 1000
... uniformly on each compact subset on the half - plane ( z ) > 0. If { f } were known to be uniformly convergent in a neighborhood of U , the analyticity of its limit fo would be clear . Unfortunately it is not clear that the sequence f ...
... uniformly on each compact subset on the half - plane ( z ) > 0. If { f } were known to be uniformly convergent in a neighborhood of U , the analyticity of its limit fo would be clear . Unfortunately it is not clear that the sequence f ...
Page 1001
... uniformly on any portion of Q whose closure contains neither a nor b . Let M be a bound for the sequence yn so that ... uniformly on Q to the function g given by the equations g ( z ) = ƒμ ( z ) ( ≈ — a ) 2 ( z — b ) 2 , = : 0 za , b ...
... uniformly on any portion of Q whose closure contains neither a nor b . Let M be a bound for the sequence yn so that ... uniformly on Q to the function g given by the equations g ( z ) = ƒμ ( z ) ( ≈ — a ) 2 ( z — b ) 2 , = : 0 za , b ...
Page 1108
... uniformly in i , i = r ( ii ) lim , Σ , 2 , ( m ) | * = 0 uniformly in m , it follows that i = 1 ( iii ) Σ12 , ( m ) is bounded uniformly in m , ( iv ) 2 ( m ) → 0 as i∞o uniformly in m . Thus , by the above inequality , ∞ lim II ...
... uniformly in i , i = r ( ii ) lim , Σ , 2 , ( m ) | * = 0 uniformly in m , it follows that i = 1 ( iii ) Σ12 , ( m ) is bounded uniformly in m , ( iv ) 2 ( m ) → 0 as i∞o uniformly in m . Thus , by the above inequality , ∞ lim II ...
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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