Linear Operators, Part 2 |
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Page 1094
... inequality by the elementary inequality \ x \ ” + | y | P ≥ \ x + y | ” , valid in this range of p . Similarly , using Corollary 3 , and the Hölder inequality { Σa‚ß ‚ \ " } 1 / " ≤ { Σ \ α , " } 1 / ~ { Σß , \ " a } 1 / ra , valid if ...
... inequality by the elementary inequality \ x \ ” + | y | P ≥ \ x + y | ” , valid in this range of p . Similarly , using Corollary 3 , and the Hölder inequality { Σa‚ß ‚ \ " } 1 / " ≤ { Σ \ α , " } 1 / ~ { Σß , \ " a } 1 / ra , valid if ...
Page 1105
... inequality ( a ) for operators in a d - dimensional Hilbert space . Arguing as in the paragraphs of the proof of Lemma 14 following formula ( 3 ) of that proof , where we proved a bilinear inequality quite similar to our present ...
... inequality ( a ) for operators in a d - dimensional Hilbert space . Arguing as in the paragraphs of the proof of Lemma 14 following formula ( 3 ) of that proof , where we proved a bilinear inequality quite similar to our present ...
Page 1774
... inequality , known as the Schwarz inequality , will be proved first . It follows from the postulates for § that the Schwarz inequality is valid if either a or y is zero . Hence suppose that x0y . For an arbitrary complex number a where ...
... inequality , known as the Schwarz inequality , will be proved first . It follows from the postulates for § that the Schwarz inequality is valid if either a or y is zero . Hence suppose that x0y . For an arbitrary complex number a where ...
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 domain 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 linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma 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