Linear Operators, Part 2 |
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Page 1050
... function f defined on Euclidean n - space E " , supposing that ƒ has a finite number of " singularities " at which it is not Lebesgue integrable , and defining a certain Cauchy - type principal value integral for f 1 DEFINITION . Let f ...
... function f defined on Euclidean n - space E " , supposing that ƒ has a finite number of " singularities " at which it is not Lebesgue integrable , and defining a certain Cauchy - type principal value integral for f 1 DEFINITION . Let f ...
Page 1196
... definition is as follows . 5 DEFINITION . Let E be the resolution of the identity for the self adjoint operator T and let ƒ be a complex Borel function defined E - almost everywhere on the real axis . Then the operator f ( T ) is defined ...
... definition is as follows . 5 DEFINITION . Let E be the resolution of the identity for the self adjoint operator T and let ƒ be a complex Borel function defined E - almost everywhere on the real axis . Then the operator f ( T ) is defined ...
Page 1645
... functions f such that auf belongs to La ( E2 ) . " In order for such an answer to make sense , it is desirable that we should be able to define a v for every function , differentiable or not , and irrespective of whether af belongs to ...
... functions f such that auf belongs to La ( E2 ) . " In order for such an answer to make sense , it is desirable that we should be able to define a v for every function , differentiable or not , and irrespective of whether af belongs to ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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Common terms and phrases
adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero