Linear Operators: Spectral theory |
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Page 1051
... evident from Definition 1. Statements ( ii ) and ( iii ) are evident consequences of Definition 1 and of the formulae √gn ¥ ( x ) dx = √gn ¥ ( Ux ) dx , √ gn ¥ ( xx ) dx = | x | ̄ ” [ gn ¥ ( x ) dx , which are valid for every ...
... evident from Definition 1. Statements ( ii ) and ( iii ) are evident consequences of Definition 1 and of the formulae √gn ¥ ( x ) dx = √gn ¥ ( Ux ) dx , √ gn ¥ ( xx ) dx = | x | ̄ ” [ gn ¥ ( x ) dx , which are valid for every ...
Page 1173
... evident manner be identified with the L - space on the product set E " S and the inequality to be proved reduces to ( 41 ) SS S SJ En Q ( x - y ) En x - yn fly , s ) dy dxds ≤ cff · ƒ_ƒ__ \ f ( x , s ) \ ' dxds . En This inequality is ...
... evident manner be identified with the L - space on the product set E " S and the inequality to be proved reduces to ( 41 ) SS S SJ En Q ( x - y ) En x - yn fly , s ) dy dxds ≤ cff · ƒ_ƒ__ \ f ( x , s ) \ ' dxds . En This inequality is ...
Page 1661
... evident , and instead of giving detailed statements which apply to our new context we shall simply refer to " Lemma 6 , " etc. , as " generalized to D „ ( I ) " when necessary . Lemma 10 applies similarly in an evident way to the new ...
... evident , and instead of giving detailed statements which apply to our new context we shall simply refer to " Lemma 6 , " etc. , as " generalized to D „ ( I ) " when necessary . Lemma 10 applies similarly in an evident way to the new ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
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Common terms and phrases
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