Linear Operators: Spectral theory |
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Page 1064
... satisfies the inequality g , IA , f ,, where I = Ss 2 ( w ) u ( dw ) . To do this , let { m } be a sequence of odd functions , each infinitely often differentiable in the neighborhood of the unit sphere , such that 2m ( tx ) = 2m ( x ) ...
... satisfies the inequality g , IA , f ,, where I = Ss 2 ( w ) u ( dw ) . To do this , let { m } be a sequence of odd functions , each infinitely often differentiable in the neighborhood of the unit sphere , such that 2m ( tx ) = 2m ( x ) ...
Page 1164
... satisfies a suitable , rather weak , continuity hypothesis , then the singular integral q ( x ) = S Q ( x - y ) f ... satisfies Alp ( x ) | 1 - ε dx < ∞ if ƒ e L1 ( E " ) , ɛ > 0 , and A is bounded ; ( iii ) satisfies { √ ( x ) \ dx } ...
... satisfies a suitable , rather weak , continuity hypothesis , then the singular integral q ( x ) = S Q ( x - y ) f ... satisfies Alp ( x ) | 1 - ε dx < ∞ if ƒ e L1 ( E " ) , ɛ > 0 , and A is bounded ; ( iii ) satisfies { √ ( x ) \ dx } ...
Page 1602
... satisfies S ' ' If ( s ) \ 2 ds = O ( tk ) for some k > 0. Then the point λ belongs to the essential spectrum of T ( Wintner [ 17 ] ) . ( 49 ) Suppose that the function q is bounded below , and that for some constant k > 0 every ...
... satisfies S ' ' If ( s ) \ 2 ds = O ( tk ) for some k > 0. Then the point λ belongs to the essential spectrum of T ( Wintner [ 17 ] ) . ( 49 ) Suppose that the function q is bounded below , and that for some constant k > 0 every ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero